Method of differences with partial fractionsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $S_n=\sum_{r=1}^{n}\frac{1}{r(r+1)}$.

   (a)

   Which expression is equal to $\frac{1}{r(r+1)}$?

   [1 mark]

   • A$\frac{1}{r}+\frac{1}{r+1}$
   • B$\frac{1}{r}-\frac{1}{r+1}$
   • C$\frac{1}{r+1}-\frac{1}{r}$
   • D$\frac{1}{r^2}-\frac{1}{r+1}$

   (b)

   Which expression is equal to $S_n$?

   [1 mark]

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

   (c)

   Find the least value of $n$ for which $S_n>0.99$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $u_r=r^3-(r-1)^3$ for positive integers $r$.

   (a)

   Which expression is equal to $u_r$?

   [1 mark]

   • A$3r^2+3r+1$
   • B$3r^2-3r-1$
   • C$3r^2-3r$
   • D$3r^2-3r+1$

   (b)

   Which expression is equal to $\sum_{r=1}^{n}u_r$?

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $u_r=\frac{1}{(2r-1)(2r+1)}$ and $S_n=\sum_{r=1}^{n}u_r$.

   (a)

   Express $u_r$ in partial fractions and hence show that $S_n=\frac{n}{2n+1}$.

   [3 marks]

   (b)

   Hence find $\sum_{r=n+1}^{2n}u_r$ in terms of $n$, giving your answer as a single fraction in its simplest form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $u_r=\frac{1}{r(r+2)}$ and $S_n=\sum_{r=1}^{n}u_r$.

   (a)

   (i) Show that $u_r=\frac12\left(\frac1r-\frac{1}{r+2}\right)$.
   (ii) Hence show that $S_n=\frac34-\frac{2n+3}{2(n+1)(n+2)}$.
   (iii) Deduce the value of $\sum_{r=1}^{\infty}u_r$.

   [6 marks]

   (b)

   (i) Find the exact value of $\sum_{r=11}^{\infty}u_r$.
   (ii) Find the smallest value of $n$ for which $\frac34-S_n<0.01$.

   [6 marks]

   Total for question 4: 12 marks

End of questions