19N.2.SL.TZ0.S_5

pestleMathematicsAASLPaper 219N· sl-1-3-geometric-sequences-and-seriessource ↗

The first two terms of a geometric sequence are ${u_1} = 2.1$ and ${u_2} = 2.226$ .

Find the value of $r$ .

[2]

Find the value of ${u_{10}}$ .

[2]

Find the least value of $n$ such that ${S_n} > 5543$ .

[3]

Markscheme / solution

valid approach (M1)

eg $\frac{{{u_1}}}{{{u_2}}}$ , $\frac{{2.226}}{{2.1}}$ , $2.226 = 2.1r$

$r = 1.06$ (exact) A1 N2

[2 marks]

correct substitution (A1)

eg $2.1 \times {1.06^9}$

$3.54790$ A1 N2

${u_{10}} = 3.55$

[2 marks]

correct substitution into ${S_n}$ formula (A1)

eg $\frac{{2.1\left( {{{1.06}^n} - 1} \right)}}{{1.06 - 1}}$ , $\frac{{2.1\left( {{{1.06}^n} - 1} \right)}}{{1.06 - 1}} > 5543$ , $2.1\left( {{{1.06}^n} - 1} \right) = 332.58$ , sketch of ${S_n}$ and $y = 5543$

correct inequality for $n$ or crossover values A1

eg $n > 87.0316$ , ${S_{87}} = 5532.73$ and ${S_{88}} = 5866.79$

$n = 88$ A1 N2

[3 marks]

Examiners’ report

[N/A]