A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2018 · Q710 Marks
Let S_N = Σ(from r=1 to N) (3r+1)(3r+4) and T_N = Σ(from r=1 to N) 1/((3r+1)(3r+4)).
- (i)Use standard results from the List of Formulae (MF10) to show that S_N = N(3N² + 12N + 13).[3]
- (ii)Use the method of differences to show that T_N = 1/12 - 1/(3(3N + 4)).[3]
- (iii)Deduce that S_N/T_N is an integer.[2]
- (iv)Find lim_(N→∞) S_N/(N³T_N).[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.
- [M1] Expands
- [M1] Substitutes formulae for Σr and Σr².
- [A1] Shows simplification to the given answer (AG).
- [B1] Finds partial fractions.
- [M1] Expresses terms as differences.
- [A1] Cancels to given answer (AG).
- [M1] Writes S_N/T_N as a polynomial
- [A1] Justifies expression being integer
- [M1] Divides expression in (iii) by N³ and takes limit
- [A1]
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