A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2013 · Q37 Marks
It is given that S_n = ∑_{r=1}^n u_r = 2n² + n. Write down the values of S1, S2, S3, S4. Express u_r in terms of r, justifying your answer. Find ∑_{r=n+1}^{2n} u_r.
- (a)Write down the values of S1, S2, S3, S4. Express u_r in terms of r, justifying your answer.[4]
- (b)Find ∑_{r=n+1}^{2n} u_r.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.
- [B1] (a) Writes first four sums. S1...S4~3, 10, 21, 36
- [B1B1] Deduces first four terms, conjectures and justifies result. u1...u4~3, 7, 11, 15 ⇒ ur = 4r – 1
- [B1] since Sn = n/2 {6 + 4(n-1)} = 2n² + n as given.
- [B1B1] Or ur = Sr − Sr−1 = 2r² + r − 2(r-1)² – (r-1) = 4r - 1
- [B1] (b) Obtains required sum. ∑_{r=n+1}^{2n} (4r-1)= 4.2n(2n+1)/2 - 4.n(n+1)/2 - n
- [A1] = 8n² + 2n - (2n² + n) = 6n² + n
- [M1A1] Or Sum of AP = n/2 (4n+3+8n-1)=6n² +n
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