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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 Find 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 exact value of S k .

[2]
b.

Show that F = 3240 .

[5]
c.

An infinite geometric series is given as  S = a + a 2 + a 2 + a Z + .

Find the largest value of a such that S < F .

[5]
d.

Markscheme

correct substitution      (A1)

eg    300 2 ( 1.3 + 31.2 ) 300 2 [ 2 ( 1.3 ) + ( 300 1 ) ( 0.1 ) ] 300 2 [ 2.6 + 299 ( 0.1 ) ]  

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 × 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 +   (accept  F = u 3 + u 6 + u 9 + )

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

100 2 ( 1.5 + 31.2 ) ,   50 ( 2 × 1.5 + 99 × 0.3 ) ,  1635

valid approach (seen anywhere)       (M1)

eg     S k ( u 3 + u 6 + ) ,   S k 100 2 ( 1.5 + 31.2 ) S k  (their sum for  ( u 3 + u 6 + ) )

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    1 2 ,  0.707106… ,   a 0.293

correct working (accept equation)        (A1)

eg    a 1 1 2 < 3240

correct working     A1

 

METHOD 1 (analytical)

eg    3240 × ( 1 1 2 ) ,   a < 948.974 ,  948.974

METHOD 2 (using table, must find both S values)

eg   when  a = 948 ,   S = 3236.67   AND  when  a = 949 ,   S = 3240.08

a = 948        A1  N2

[5 marks]

d.

Examiners report

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b.
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c.
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d.

Syllabus sections

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