Question 0521

Sigma Notation
2021 Paper 1 Question 5 Variant

Question

(a)
Express
x+5(x+1)(x+2)(x+3)\frac{x + 5}{(x + 1)(x + 2)(x + 3)}
in partial fractions.
[3]
(b)
Hence find, in terms of n,{n,}
r=1nr+5(r+1)(r+2)(r+3).\sum_{r=1}^n \frac{r + 5}{(r + 1)(r + 2)(r + 3)}.
[3]
(c)
State the value of
r=1r+5(r+1)(r+2)(r+3).\sum_{r=1}^\infty \frac{r + 5}{(r + 1)(r + 2)(r + 3)}.
[1]

Answer