Method of differencesEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $u_r=\frac{1}{r(r+1)}$.

   (a)

   Which expression is equal to $u_r$?

   [1 mark]

   • A$\frac1r-\frac1{r+1}$
   • B$\frac1{r+1}-\frac1r$
   • C$\frac1r+\frac1{r+1}$
   • D$\frac1r-\frac1{r+2}$

   (b)

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

   [1 mark]

   • A$\frac1{n+1}$
   • B$\frac{n}{n+1}$
   • C$1-\frac1n$
   • D$\frac{n+1}{n+2}$

   (c)

   Find the value of $\sum_{r=10}^{19}u_r$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For all values of $r$, $(r+1)^3-r^3=3r^2+3r+1$.

   (a)

   Find $\sum_{r=1}^{n}\left[(r+1)^3-r^3\right]$.

   [1 mark]

   • A$n^3$
   • B$(n+1)^3$
   • C$n^3-1$
   • D$(n+1)^3-1$

   (b)

   Find the value of $\sum_{r=1}^{5}\left(3r^2+3r+1\right)$.

   [1 mark]

   • A$216$
   • B$124$
   • C$215$
   • D$125$

   (c)

   Hence find the value of $\sum_{r=11}^{20}\left(3r^2+3r+1\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $u_r=\frac{1}{(r+1)(r+3)}$.

   (a)

   Express $u_r$ in the form $\frac{A}{r+1}+\frac{B}{r+3}$, where $A$ and $B$ are constants to be found.

   [3 marks]

   (b)

   Hence show that $\sum_{r=1}^{n}u_r=\frac5{12}-\frac{2n+5}{2(n+2)(n+3)}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two sequences have $r$th terms $u_r=\frac{2r+1}{r^2(r+1)^2}$ and $v_r=\frac{1}{r(r+1)(r+2)}$.

   (a)

   (i) Show that $u_r=\frac1{r^2}-\frac1{(r+1)^2}$.
   (ii) Hence find $\sum_{r=1}^{n}u_r$, giving your answer as a single fraction.
   (iii) Deduce the value of $\sum_{r=1}^{\infty}u_r$.

   [6 marks]

   (b)

   Using partial fractions, show that $\sum_{r=1}^{n}v_r=\frac{n(n+3)}{4(n+1)(n+2)}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions