IB Mathematics Applications & Interpretation Topic AHL 1.11—sum of Infinite Geometric Sequences Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 1.11—sum of Infinite Geometric Sequences Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 1.11—sum of Infinite Geometric Sequences with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 1.11—sum of Infinite Geometric Sequences and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

For a geometric progression, the third term is $6$ and the sixth term is $\frac{2}{9}$ .

Determine the common ratio for this progression.

[3]

Calculate the sum to infinity for this specific progression.

[2]

Question 2

HLPaper 2

An arithmetic progression has terms $v_{1} = 1.8$ , $v_{2} = 2.1$ , and a specific term $v_{k} = 64.8$ . Consider the terms $v_{n}$ where $n \leq k$ . Let $R$ be the total sum of all terms $v_{n}$ for which $n$ is not a multiple of $4$ .

Calculate the value of $k$ .

[4]

Determine the total sum of the sequence from $n = 1$ to $k$ .

[2]

Show that $R = 5294.7$ .

[5]

An infinite geometric sequence has sum to infinity $S_{\infty} = p + \frac{p}{2} + \frac{p}{4} + \dots$ , where $p \in Z^{+}$ . Find the greatest integer value of $p$ such that $S_{\infty} < R$ .

[5]

Question 3

HLPaper 1

Consider an infinite geometric series whose first two terms are $4 lo g_{2} y, lo g_{2} y$ , where $y > 0$ . An arithmetic sequence begins with $lo g_{2} y, lo g_{2} (\frac{y}{2}), lo g_{2} (\frac{y}{4})$ , where $y > 0$ . Let $S_{12}$ be the sum of the first 12 terms of the arithmetic sequence.

Determine the common ratio $r$ of the geometric sequence.

[2]

Demonstrate that the sum to infinity of this geometric sequence is $\frac{16}{3} lo g_{2} y$ .

[2]

Calculate the common difference $d$ of the arithmetic sequence as an integer value.

[4]

Verify that $S_{12} = 12 lo g_{2} y - 66$ .

[2]

If $S_{12}$ equals exactly $\frac{3}{4}$ of the sum to infinity of the geometric sequence, solve for $y$ in the form $2^{p}$ , where $p \in Q$ .

[3]

Question 4

HLPaper 2

An arithmetic progression has terms $u_{1} = 2.5$ , $u_{2} = 2.9$ , and a specific term $u_{m} = 74.1$ . Consider the terms $u_{n}$ where $n \leq m$ . Let $T$ be the total sum of all terms $u_{n}$ for which $n$ is not a multiple of $5$ .

Calculate the value of $m$ .

[4]

Determine the total sum of the sequence from $n = 1$ to $m$ .

[2]

Show that $T = 5486.4$ .

[5]

An infinite geometric sequence has sum to infinity $S_{\infty} = p + \frac{p}{3} + \frac{p}{9} + \dots$ , where $p \in Z^{+}$ . Find the greatest integer value of $p$ such that $S_{\infty} < T$ .

[5]

Question 5

HLPaper 2

A decorative strip is made from infinitely many right-angled isosceles triangular tiles placed consecutively along the edge of a shelf. In each arrangement, one equal side of each tile lies on the shelf edge, and the tiles do not overlap, so the total shelf edge covered is the sum of these side lengths.

In one arrangement, the side lengths on the shelf edge are $r$ cm, $r^{2}$ cm, $r^{3}$ cm, $\dots$ , where $0 < r < 1$ , and the pattern continues indefinitely. The strip covers exactly $4$ cm of the shelf edge. Demonstrate that $r = \frac{4}{5}$ .

[5]

In a second arrangement, the side lengths on the shelf edge are $s$ cm, $s^{2}$ cm, $s^{3}$ cm, $\dots$ , where $0 < s < 1$ . If the combined area of all the tiles is $\frac{2}{5} cm^{2}$ , calculate the total shelf-edge length $L$ .

[9]

IB Mathematics Applications & Interpretation AHL 1.11—sum of Infinite Geometric Sequences Questionbank

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IB Mathematics Applications & Interpretation AHL 1.11—sum of Infinite Geometric Sequences Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.11—sum of Infinite Geometric Sequences Questionbank

Unlock all 27 questions with a free account