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Date November 2016 Marks available 1 Reference code 16N.1.hl.TZ0.6
Level HL only Paper 1 Time zone TZ0
Command term Write down Question number 6 Adapted from N/A

Question

The sum of the first \(n\) terms of a sequence \(\{ {u_n}\} \) is given by \({S_n} = 3{n^2} - 2n\), where \(n \in {\mathbb{Z}^ + }\).

Write down the value of \({u_1}\).

[1]
a.

Find the value of \({u_6}\).

[2]
b.

Prove that \(\{ {u_n}\} \) is an arithmetic sequence, stating clearly its common difference.

[4]
c.

Markscheme

\({u_1} = 1\)    A1

[1 mark]

a.

\({u_6} = {S_6} - {S_5} = 31\)    M1A1

[2 marks]

b.

\({u_n} = {S_n} - {S_{n - 1}}\)    M1

\( = (3{n^2} - 2n) - \left( {3{{(n - 1)}^2} - 2(n - 1)} \right)\)

\( = (3{n^2} - 2n) - (3{n^2} - 6n + 3 - 2n + 2)\)

\( = 6n - 5\)    A1

\(d = {u_{n + 1}} - {u_n}\)    R1

\( = 6n + 6 - 5 - 6n + 5\)

\( = \left( {6(n + 1) - 5} \right) - (6n - 5)\)

\( = 6\) (constant)     A1

 

Notes: Award R1 only if candidate provides a clear argument that proves that the difference between ANY two consecutive terms of the sequence is constant. Do not accept examples involving particular terms of the sequence nor circular reasoning arguments (eg use of formulas of APs to prove that it is an AP). Last A1 is independent of R1.

 

[4 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
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c.

Syllabus sections

Topic 1 - Core: Algebra » 1.1 » Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.
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