Worksheet · Calculus II · Unit 4A

Geometric Series Worksheet with Answers and Printable PDF

Twenty finite, infinite, shifted-index, decimal, modeling, and error-analysis problems.

20 problems60 min estimated timeFoundational to intermediate progression

What is included

Practice geometric sequences and series with complete HTML solutions and separate student and key PDFs.

Skills assessed

common ratio
finite sums
infinite sums
index shifts
applied models

Prerequisites

sequences
sigma notation

Geometric Series Worksheet instructional sequence — Twenty finite, infinite, shifted-index, decimal, modeling, and error-analysis problems. The numbered labels and written sequence preserve meaning without relying on color.

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF SVG PNG

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Printable preview

Find the common ratio of 3, 12, 48, …
Find a_8 when a_1=5 and r=2.
Evaluate $\sum_{k=0}^{5}3(2)^k$ .
Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .
Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .
Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .
Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.
Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .
Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.
Write 0.333… as a fraction.
Write 0.272727… as a fraction.
Write 0.145145… as a fraction.
A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.
Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.
A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.
Find S_n for 7+7(0.8)+7(0.8)^2+…
An infinite geometric series has first term 6 and sum 15. Find r.
An infinite geometric series has ratio -1/4 and sum 8. Find its first term.
A student uses a/(1-r) for r=2. Diagnose the error.
Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Find the common ratio of 3, 12, 48, …

Answer: $r=4$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $r=4$ .

Problem 2: Find a_8 when a_1=5 and r=2.

Answer: $a_8=640$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $a_8=640$ .

Problem 3: Evaluate $\sum_{k=0}^{5}3(2)^k$ .

Answer: $189$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $189$ .

Problem 4: Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .

Answer: $31$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $31$ .

Problem 5: Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .

Answer: $18$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $18$ .

Problem 6: Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .

Answer: $16/3$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $16/3$ .

Problem 7: Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.

Answer: $diverges$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $diverges$ .

Problem 8: Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .

Answer: $1/24$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1/24$ .

Problem 9: Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.

Answer: $3/2$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/2$ .

Problem 10: Write 0.333… as a fraction.

Answer: $1/3$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1/3$ .

Problem 11: Write 0.272727… as a fraction.

Answer: $3/11$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/11$ .

Problem 12: Write 0.145145… as a fraction.

Answer: $145/999$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $145/999$ .

Problem 13: A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.

Answer: $170/3\text{ m}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $170/3\text{ m}$ .

Problem 14: Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.

Answer: $431.01\text{ dollars}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $431.01\text{ dollars}$ .

Problem 15: A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.

Answer: $1$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1$ .

Problem 16: Find S_n for 7+7(0.8)+7(0.8)^2+…

Answer: $35(1-0.8^n)$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $35(1-0.8^n)$ .

Problem 17: An infinite geometric series has first term 6 and sum 15. Find r.

Answer: $3/5$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/5$ .

Problem 18: An infinite geometric series has ratio -1/4 and sum 8. Find its first term.

Answer: $10$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $10$ .

Problem 19: A student uses a/(1-r) for r=2. Diagnose the error.

Answer: $|r|\ge1,\text{ so the infinite series diverges}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $|r|\ge1,\text{ so the infinite series diverges}$ .

Problem 20: Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .

Answer: $605$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $605$ .

Common errors

Using the infinite formula when |r|≥1.
Mistaking the first listed term for the coefficient a after an index shift.
Forgetting that a repeating-decimal series begins at a decimal place.

Worksheet · Calculus II · Unit 4A

Geometric Series Worksheet with Answers and Printable PDF

Twenty finite, infinite, shifted-index, decimal, modeling, and error-analysis problems.

20 problems60 min estimated timeFoundational to intermediate progression

What is included

Practice geometric sequences and series with complete HTML solutions and separate student and key PDFs.

Skills assessed

common ratio
finite sums
infinite sums
index shifts
applied models

Prerequisites

sequences
sigma notation

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF SVG PNG

Device-only study control

Ready for an honest first attempt?

This control sends only the resource id and type—never your answers or identity.

Printable preview

Find the common ratio of 3, 12, 48, …
Find a_8 when a_1=5 and r=2.
Evaluate $\sum_{k=0}^{5}3(2)^k$ .
Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .
Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .
Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .
Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.
Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .
Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.
Write 0.333… as a fraction.
Write 0.272727… as a fraction.
Write 0.145145… as a fraction.
A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.
Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.
A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.
Find S_n for 7+7(0.8)+7(0.8)^2+…
An infinite geometric series has first term 6 and sum 15. Find r.
An infinite geometric series has ratio -1/4 and sum 8. Find its first term.
A student uses a/(1-r) for r=2. Diagnose the error.
Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Find the common ratio of 3, 12, 48, …

Answer: $r=4$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $r=4$ .

Problem 2: Find a_8 when a_1=5 and r=2.

Answer: $a_8=640$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $a_8=640$ .

Problem 3: Evaluate $\sum_{k=0}^{5}3(2)^k$ .

Answer: $189$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $189$ .

Problem 4: Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .

Answer: $31$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $31$ .

Problem 5: Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .

Answer: $18$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $18$ .

Problem 6: Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .

Answer: $16/3$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $16/3$ .

Problem 7: Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.

Answer: $diverges$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $diverges$ .

Problem 8: Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .

Answer: $1/24$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1/24$ .

Problem 9: Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.

Answer: $3/2$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/2$ .

Problem 10: Write 0.333… as a fraction.

Answer: $1/3$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1/3$ .

Problem 11: Write 0.272727… as a fraction.

Answer: $3/11$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/11$ .

Problem 12: Write 0.145145… as a fraction.

Answer: $145/999$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $145/999$ .

Problem 13: A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.

Answer: $170/3\text{ m}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $170/3\text{ m}$ .

Problem 14: Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.

Answer: $431.01\text{ dollars}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $431.01\text{ dollars}$ .

Problem 15: A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.

Answer: $1$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $1$ .

Problem 16: Find S_n for 7+7(0.8)+7(0.8)^2+…

Answer: $35(1-0.8^n)$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $35(1-0.8^n)$ .

Problem 17: An infinite geometric series has first term 6 and sum 15. Find r.

Answer: $3/5$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $3/5$ .

Problem 18: An infinite geometric series has ratio -1/4 and sum 8. Find its first term.

Answer: $10$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $10$ .

Problem 19: A student uses a/(1-r) for r=2. Diagnose the error.

Answer: $|r|\ge1,\text{ so the infinite series diverges}$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $|r|\ge1,\text{ so the infinite series diverges}$ .

Problem 20: Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .

Answer: $605$

Why this method: Geometric-series structure matches the mathematical structure before any algebraic cleanup.

Identify the first included term, common ratio, and whether the sum is finite or infinite.
Use $S_n=a(1-r^n)/(1-r)$ for a finite sum or $S=a/(1-r)$ only when $|r|<1$ .
Substitution and simplification give $605$ .

Common errors

Using the infinite formula when |r|≥1.
Mistaking the first listed term for the coefficient a after an index shift.
Forgetting that a repeating-decimal series begins at a decimal place.

What is included

Skills assessed

Prerequisites

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Find the common ratio of 3, 12, 48, …

Problem 2: Find a_8 when a_1=5 and r=2.

Problem 3: Evaluate ∑k=053(2)k\sum_{k=0}^{5}3(2)^k∑k=05​3(2)k.

Problem 4: Evaluate ∑k=0416(1/2)k\sum_{k=0}^{4}16(1/2)^k∑k=04​16(1/2)k.

Problem 5: Evaluate ∑k=0∞12(1/3)k\sum_{k=0}^{\infty}12(1/3)^k∑k=0∞​12(1/3)k.

Problem 6: Evaluate ∑k=0∞8(−1/2)k\sum_{k=0}^{\infty}8(-1/2)^k∑k=0∞​8(−1/2)k.

Problem 7: Decide whether ∑k=0∞5(1.02)k\sum_{k=0}^{\infty}5(1.02)^k∑k=0∞​5(1.02)k converges.

Problem 8: Evaluate ∑n=3∞2(1/4)n\sum_{n=3}^{\infty}2(1/4)^n∑n=3∞​2(1/4)n.

Problem 9: Rewrite ∑n=1∞31−n\sum_{n=1}^{\infty}3^{1-n}∑n=1∞​31−n in arkar^kark form and sum it.

Problem 10: Write 0.333… as a fraction.

Problem 11: Write 0.272727… as a fraction.

Problem 12: Write 0.145145… as a fraction.

Problem 13: A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.

Problem 14: Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.

Problem 15: A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.

Problem 16: Find S_n for 7+7(0.8)+7(0.8)^2+…

Problem 17: An infinite geometric series has first term 6 and sum 15. Find r.

Problem 18: An infinite geometric series has ratio -1/4 and sum 8. Find its first term.

Problem 19: A student uses a/(1-r) for r=2. Diagnose the error.

Problem 20: Evaluate ∑n=265(3)n−2\sum_{n=2}^{6}5(3)^{n-2}∑n=26​5(3)n−2.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Find the common ratio of 3, 12, 48, …

Problem 2: Find a_8 when a_1=5 and r=2.

Problem 3: Evaluate ∑k=053(2)k\sum_{k=0}^{5}3(2)^k∑k=05​3(2)k.

Problem 4: Evaluate ∑k=0416(1/2)k\sum_{k=0}^{4}16(1/2)^k∑k=04​16(1/2)k.

Problem 5: Evaluate ∑k=0∞12(1/3)k\sum_{k=0}^{\infty}12(1/3)^k∑k=0∞​12(1/3)k.

Problem 6: Evaluate ∑k=0∞8(−1/2)k\sum_{k=0}^{\infty}8(-1/2)^k∑k=0∞​8(−1/2)k.

Problem 7: Decide whether ∑k=0∞5(1.02)k\sum_{k=0}^{\infty}5(1.02)^k∑k=0∞​5(1.02)k converges.

Problem 8: Evaluate ∑n=3∞2(1/4)n\sum_{n=3}^{\infty}2(1/4)^n∑n=3∞​2(1/4)n.

Problem 9: Rewrite ∑n=1∞31−n\sum_{n=1}^{\infty}3^{1-n}∑n=1∞​31−n in arkar^kark form and sum it.

Problem 10: Write 0.333… as a fraction.

Problem 11: Write 0.272727… as a fraction.

Problem 12: Write 0.145145… as a fraction.

Problem 13: A ball rebounds 70% of each previous height after a 10 m drop. Find total vertical distance.

Problem 14: Deposit 100 dollars at the end of each year for four years at 5 percent. Find the value immediately after the fourth deposit.

Problem 15: A square has area 1; each stage shades one fourth of the remaining area. Find total shaded area.

Problem 16: Find S_n for 7+7(0.8)+7(0.8)^2+…

Problem 17: An infinite geometric series has first term 6 and sum 15. Find r.

Problem 18: An infinite geometric series has ratio -1/4 and sum 8. Find its first term.

Problem 19: A student uses a/(1-r) for r=2. Diagnose the error.

Problem 20: Evaluate ∑n=265(3)n−2\sum_{n=2}^{6}5(3)^{n-2}∑n=26​5(3)n−2.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 3: Evaluate $\sum_{k=0}^{5}3(2)^k$ .

Problem 4: Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .

Problem 5: Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .

Problem 6: Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .

Problem 7: Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.

Problem 8: Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .

Problem 9: Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.

Problem 20: Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .

Problem 3: Evaluate $\sum_{k=0}^{5}3(2)^k$ .

Problem 4: Evaluate $\sum_{k=0}^{4}16(1/2)^k$ .

Problem 5: Evaluate $\sum_{k=0}^{\infty}12(1/3)^k$ .

Problem 6: Evaluate $\sum_{k=0}^{\infty}8(-1/2)^k$ .

Problem 7: Decide whether $\sum_{k=0}^{\infty}5(1.02)^k$ converges.

Problem 8: Evaluate $\sum_{n=3}^{\infty}2(1/4)^n$ .

Problem 9: Rewrite $\sum_{n=1}^{\infty}3^{1-n}$ in $ar^k$ form and sum it.

Problem 20: Evaluate $\sum_{n=2}^{6}5(3)^{n-2}$ .