AHL 1.11—Sum of infinite geometric sequences - IB Questionbank | RevisionDojo

AHL 1.11—Sum of infinite geometric sequences Questionbank

Practice AHL 1.11—Sum of infinite geometric sequences with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

In a geometric sequence, the sum to infinity is 20 and the second term is 6.

Find the first term and the common ratio of this sequence.

[5]

Question 2

HLPaper 2

A geometric sequence has first term $a=48$ and common ratio $r=0.75$ .

Show that this sequence is convergent by stating the condition for $r$ that must be satisfied.

[1]

Find the sum to infinity of this geometric sequence.

[2]

Question 3

HLPaper 2

An infinite geometric series has first term $v_{1} = b$ and second term $v_{2} = \frac{1}{4} b^{2} - 3 b$ ,where $b > 0$ .

Find the common ratio in terms of $b$ .

[2]

Find the values of $b$ for which the sum to infinity of the series exists.

[3]

Find the value of $b$ when $T_{\infty} = 76$ .

[3]

Question 4

HLPaper 2

In a bank account, interest is compounded monthly at a rate of 1% per month. An initial deposit of $P$ dollars is made.

Write an expression for the amount of money in the account after $n$ months in terms of $P$ .

[1]

If monthly withdrawals of $500$ are made starting from the end of the first month, write an expression for the sum of all withdrawals after $n$ months.

[3]

Question 5

HLPaper 2

Suppose that ${u_1}$ is the first term of a geometric series with common ratio $r$ .

Prove, by mathematical induction, that the sum of the first $n$ terms, ${s_n}$ is given by

${s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}$ , where $n \in {\mathbb{Z}^ + }$ .

Question 6

HLPaper 2

A geometric sequence has first term $a=5$ and common ratio $r=-0.2$ .

Calculate the sum to infinity of this sequence, giving your answer as a fraction.

[3]

Question 7

HLPaper 2

Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.

Find the sum to infinity of this sequence.

Find the common ratio of this sequence.

Question 8

HLPaper 2

A geometric sequence has first term $a=48$ and common ratio $r=0.75$ .

Show that this sequence is converging.

[1]

Find the sum to infinity of this geometric sequence.

[3]

Question 9

SLPaper 2

The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.

Find the common ratio.

[2]

Find the sum of the first 8 terms.

[2]

Find the least value of n for which S_n > 163.

[3]

Question 10

HLPaper 2

A ball is dropped from a height of 10 meters and bounces back to 60% of its previous height after each bounce. Calculate the total distance traveled by the ball until it comes to rest.

Calculate the total distance traveled by the ball until it comes to rest.

[5]

How many bounces are needed until the ball reaches rest?

[2]

Paper

Level

Question 1

HLPaper 2

In a geometric sequence, the sum to infinity is 20 and the second term is 6.

Find the first term and the common ratio of this sequence.

[5]

Question 2

HLPaper 2

A geometric sequence has first term $a=48$ and common ratio $r=0.75$ .

Show that this sequence is convergent by stating the condition for $r$ that must be satisfied.

[1]

Find the sum to infinity of this geometric sequence.

[2]

Question 3

HLPaper 2

An infinite geometric series has first term $v_{1} = b$ and second term $v_{2} = \frac{1}{4} b^{2} - 3 b$ ,where $b > 0$ .

Find the common ratio in terms of $b$ .

[2]

Find the values of $b$ for which the sum to infinity of the series exists.

[3]

Find the value of $b$ when $T_{\infty} = 76$ .

[3]

Question 4

HLPaper 2

In a bank account, interest is compounded monthly at a rate of 1% per month. An initial deposit of $P$ dollars is made.

Write an expression for the amount of money in the account after $n$ months in terms of $P$ .

[1]

If monthly withdrawals of $500$ are made starting from the end of the first month, write an expression for the sum of all withdrawals after $n$ months.

[3]

Question 5

HLPaper 2

Suppose that ${u_1}$ is the first term of a geometric series with common ratio $r$ .

Prove, by mathematical induction, that the sum of the first $n$ terms, ${s_n}$ is given by

${s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}$ , where $n \in {\mathbb{Z}^ + }$ .

Question 6

HLPaper 2

A geometric sequence has first term $a=5$ and common ratio $r=-0.2$ .

Calculate the sum to infinity of this sequence, giving your answer as a fraction.

[3]

Question 7

HLPaper 2

Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.

Find the sum to infinity of this sequence.

Find the common ratio of this sequence.

Question 8

HLPaper 2

A geometric sequence has first term $a=48$ and common ratio $r=0.75$ .

Show that this sequence is converging.

[1]

Find the sum to infinity of this geometric sequence.

[3]

Question 9

SLPaper 2

The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.

Find the common ratio.

[2]

Find the sum of the first 8 terms.

[2]

Find the least value of n for which S_n > 163.

[3]

Question 10

HLPaper 2

A ball is dropped from a height of 10 meters and bounces back to 60% of its previous height after each bounce. Calculate the total distance traveled by the ball until it comes to rest.

Calculate the total distance traveled by the ball until it comes to rest.

[5]

How many bounces are needed until the ball reaches rest?

[2]