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Date May 2019 Marks available 2 Reference code 19M.2.SL.TZ1.S_7
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number S_7 Adapted from N/A

Question

The first terms of an infinite geometric sequence,  u n , are 2, 6, 18, 54, …

The first terms of a second infinite geometric sequence, v n , are 2, −6, 18, −54, …

The terms of a third sequence, w n , are defined as  w n = u n + v n .

The finite series,  k = 1 225 w k  , can also be written in the form k = 0 m 4 r k .

Write down the first three non-zero terms of  w n .

[3]
a.

Find the value of  r .

[2]
b.i.

Find the value of m .

[2]
b.ii.

Markscheme

attempt to add corresponding terms      (M1)

eg    2 + 2 , 6 + ( 6 ) , 2 ( 3 ) n 1 + 2 ( 3 ) n 1

correct value for  w 5         (A1)

eg   324

4, 36, 324 (accept 4 + 36 + 324)      A1 N3

[3 marks]

a.

valid approach     (M1)

eg   4 × r 1 = 36 ,   4 × 9 n 1

r = 9   (accept  k = 0 m 4 × 9 k m  may be incorrect)      A1 N2

[2 marks]

b.i.

recognition that 225 terms of w n consists of 113 non-zero terms    (M1)

eg   1 113 0 112 ,  113

m = 112   (accept  k = 0 1 12 4 × r k r may be incorrect)      A1 N2

[2 marks]

b.ii.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.

Syllabus sections

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

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