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 , are2, −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 .

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

attempt to add corresponding terms (M1)

eg $2+2\text{,}6+\left(-6\right)\text{,}2{\left(3\right)}^{n-1}+2{\left(-3\right)}^{n-1}$

correct value for $w_5$ (A1)

eg 324

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

[3 marks]

valid approach (M1)

eg $4\times r^1=36$,$4\times 9^{n-1}$

$r=9$ (accept$\sum _{k=0}^{m}4\times 9^k$;$m$ may be incorrect) A1 N2

[2 marks]

recognition that 225 terms of $w_n$ consists of 113 non-zero terms (M1)

eg$\sum _1^{113}$, $\sum _0^{112}$, 113

$m=112$ (accept$\sum _{k=0}^{1}124\times r^k$;$r$ may be incorrect) A1 N2

[2 marks]