A-LevelMathematicsSeries9709/12 · May/June 2015 · Q89 Marks

The first, second and last terms in an arithmetic progression are 56, 53 and –22 respectively. Find the sum of all the terms in the progression.

(a)The first, second and last terms in an arithmetic progression are 56, 53 and –22 respectively. Find the sum of all the terms in the progression.[4]
(b)(i)The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2 respectively, where k is a positive constant. Find the value of k.[3]
(b)(ii)Find the sum to infinity of the progression.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[M1] (a) -22 = 56 + (n - 1)(-3) Uses correct u_n formula.
[A1] (a) → n = 27 co Needs positive integer n
[M1] (a) S27 = (27/2)(112+26(-3))
[A1] (a) → 459 Co
[M1] (b)(i) Either (2k)/(2k+6) = (k+2)/(2k) Correct method for equation in k.
[DM1] (b)(i) → 2k² - 10k - 12 = 0 Forms quad. or cubic equation with no brackets or fractions.
[A1] (b)(i) → k = 6 Co
[M1] (b)(ii) S∞ = a/(1-r) with r = 2k/(2k+6) or 2k/(2k+6) (= 2/3) Needs attempt at a and r and S∞
[A1] (b)(ii) → 54 Co

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