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Date	May 2019	Marks available	5	Reference code	19M.2.SL.TZ2.S_10
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Show that	Question number	S_10	Adapted from	N/A

Question

In an arithmetic sequence, ${u_1} = 1.3$ , ${u_2} = 1.4$ and ${u_k} = 31.2$ .

Consider the terms, ${u_n}$ , of this sequence such that $n$ ≤ $k$ .

Let $F$ be the sum of the terms for which $n$ is not a multiple of 3.

Find the value of $k$ .

[4]

a.

Find the exact value of ${S_k}$ .

[2]

b.

Show that $F = 3240$ .

[5]

c.

An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .

[5]

d.

Markscheme

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

attempt to find $d$ (M1)

eg 1.4 − 1.3 , ${u_1} - {u_2}$ , $1.4 = 1.3 + \left( {2 - 1} \right)d$

$d = 0.1$ (may be seen in expression for ${u_n}$ ) (A1)

correct equation (A1)

eg $1.3 + \left( {k - 1} \right) \times 0.1 = 31.2$ , $0.1k = 30$

$k = 300$ A1 N3

[4 marks]

a.

correct substitution (A1)

eg $\frac{{300}}{2}\left( {1.3 + 31.2} \right)$ , $\frac{{300}}{2}\left[ {2\left( {1.3} \right) + \left( {300 - 1} \right)\left( {0.1} \right)} \right]$ , $\frac{{300}}{2}\left[ {2.6 + 299\left( {0.1} \right)} \right]$

${S_k} = 4875$ A1 N2

[2 marks]

b.

recognizing need to find the sequence of multiples of 3 (seen anywhere) (M1)

eg first term is ${u_3}$ (= 1.5) (accept notation ${u_1} = 1.5$ ) ,

$d = 0.1 \times 3$ (= 0.3) , 100 terms (accept $n = 100$ ), last term is 31.2

(accept notation ${u_{100}} = 31.2$ ) , ${u_3} + {u_6} + {u_9} + \ldots$ (accept $F = {u_3} + {u_6} + {u_9} + \ldots$ )

correct working for sum of sequence where n is a multiple of 3 A2

$\frac{{100}}{2}\left( {1.5 + 31.2} \right)$ , $50\left( {2 \times 1.5 + 99 \times 0.3} \right)$ , 1635

valid approach (seen anywhere) (M1)

eg ${S_k} - \left( {{u_3} + {u_6} + \ldots } \right)$ , ${S_k} - \frac{{100}}{2}\left( {1.5 + 31.2} \right)$ , ${S_k} -$ (their sum for $\left( {{u_3} + {u_6} + \ldots } \right)$ )

correct working (seen anywhere) A1

eg ${S_k} - 1635$ , 4875 − 1635

$F = 3240$ AG N0

[5 marks]

c.

attempt to find $r$ (M1)

eg dividing consecutive terms

correct value of $r$ (seen anywhere, including in formula)

eg $\frac{1}{{\sqrt 2 }}$ , 0.707106… , $\frac{a}{{0.293 \ldots }}$

correct working (accept equation) (A1)

eg $\frac{a}{{1 - \frac{1}{{\sqrt 2 }}}} < 3240$

correct working A1

METHOD 1 (analytical)

eg $3240 \times \left( {1 - \frac{1}{{\sqrt 2 }}} \right)$ , $a < 948.974$ , 948.974

METHOD 2 (using table, must find both ${S_\infty }$ values)

eg when $a = 948$ , ${S_\infty } = 3236.67 \ldots$ AND when $a = 949$ , ${S_\infty } = 3240.08 \ldots$

$a = 948$ A1 N2

[5 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

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

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Topic 1—Number and algebra