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Date	May 2018	Marks available	4	Reference code	18M.1.AHL.TZ2.H_5
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Show that	Question number	H_5	Adapted from	N/A

Question

The geometric sequence u₁, u₂, u₃, … has common ratio r.

Consider the sequence $A = \left\{ {{a_n} = {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|{\text{:}}\,n \in {\mathbb{Z}^ + }} \right\}$ .

Show that A is an arithmetic sequence, stating its common difference d in terms of r.

[4]

a.

A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.

[3]

b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

state that ${u_n} = {u_1}{r^{n - 1}}$ (or equivalent) A1

attempt to consider ${{a_n}}$ and use of at least one log rule M1

${\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right| = {\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right| + \left( {n - 1} \right){\text{lo}}{{\text{g}}_2}\left| r \right|$ A1

(which is an AP) with $d = {\text{lo}}{{\text{g}}_2}\left| r \right|$ (and 1^st term ${\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right|$ ) A1

so A is an arithmetic sequence AG

Note: Condone absence of modulus signs.

Note: The final A mark may be awarded independently.

Note: Consideration of the first two or three terms only will score M0.

[4 marks]

METHOD 2

consideration of $\left( {d = } \right){a_{n + 1}} - {a_n}$ M1

$\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {{u_{n + 1}}} \right| - {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|$

$\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right|$ M1

$\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| r \right|$ A1

which is constant R1

Note: Condone absence of modulus signs.

Note: The final A mark may be awarded independently.

Note: Consideration of the first two or three terms only will score M0.

a.

attempting to solve $\frac{3}{{1 - r}} = 4$ M1

$r = \frac{1}{4}$ A1

$d = - \,2$ A1

[3 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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Topic 1—Number and algebra