The method of differencesEdexcel International A Level Further Maths: Subtopic test

1. 1
   The general term of a series is $u_r=\frac{1}{r(r+2)}$ for $r\ge1$.

   (a)

   Express $u_r$ in partial fractions.

   [1 mark]

   • A$\frac1r-\frac1{r+2}$
   • B$\frac12\left(\frac1r+\frac1{r+2}\right)$
   • C$\frac12\left(\frac1r-\frac1{r+2}\right)$
   • D$2\left(\frac1r-\frac1{r+2}\right)$

   (b)

   When $\sum_{r=1}^{n}u_r$ is evaluated by the method of differences, how many terms remain after the cancelling?

   [1 mark]

   • Atwo
   • Bthree
   • C$n$
   • Dfour

   (c)

   Find $\sum_{r=1}^{\infty}u_r$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(r)=r!$ for positive integers $r$.

   (a)

   Simplify $f(r+1)-f(r)$.

   [1 mark]

   • A$r!$
   • B$r\cdot r!$
   • C$(r+1)\cdot r!$
   • D$1$

   (b)

   Hence find $\sum_{r=1}^{n}r\cdot r!$.

   [1 mark]

   • A$(n+1)!-1$
   • B$(n+1)!$
   • C$n!-1$
   • D$(n+1)!-2$

   (c)

   Hence find $\sum_{r=3}^{5}r\cdot r!$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A series has general term $u_r=\frac{1}{(2r-1)(2r+1)}$ for $r\ge1$.

   (a)

   Express $u_r$ in partial fractions.

   [3 marks]

   (b)

   Hence find $\sum_{r=1}^{n}u_r$, giving your answer as a single fraction.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The terms of a series are $u_r=\frac{1}{r(r+1)(r+2)}$ for $r\ge1$, and $S_n=\sum_{r=1}^{n}u_r$.

   (a)

   Show that $u_r=\frac12\left[\frac{1}{r(r+1)}-\frac{1}{(r+1)(r+2)}\right]$, and hence show that $S_n=\frac{n(n+3)}{4(n+1)(n+2)}$.

   [6 marks]

   (b)

   Find the least value of $n$ for which $S_n$ differs from $\sum_{r=1}^{\infty}u_r$ by less than $0.001$.

   [6 marks]

   Total for question 4: 12 marks

End of questions