19M.2.SL.TZ1.S_7

pestleMathematicsAASLPaper 219M· sl-1-2-arithmetic-sequences-and-seriessource ↗

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, $\sum\limits_{k = 1}^{225} {{w_k}}$ , can also be written in the form $\sum\limits_{k = 0}^m {4{r^k}}$ .

Write down the first three non-zero terms of ${w_n}$ .

[3]

Find the value of $r$ .

[2]

b.i.

Find the value of $m$ .

[2]

b.ii.

Markscheme / solution

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

[2 marks]

b.i.

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

eg $\sum\limits_1^{113} {}$ , $\sum\limits_0^{112} {}$ , 113

$m = 112$ (accept $\sum\limits_{k = 0}^112 {4 \times {r^k}}$ ; $r$ may be incorrect) A1 N2

[2 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.