Sums of seriesEdexcel A-Level Further Maths: Subtopic test

1. 1
   A student uses the standard summation formulae for the first $n$ positive integers, their squares and their cubes.

   (a)

   Find the value of $\sum_{r=1}^{20}r$.

   [1 mark]

   • A$190$
   • B$420$
   • C$231$
   • D$210$

   (b)

   Find the value of $\sum_{r=1}^{10}r^2$.

   [1 mark]

   • A$385$
   • B$55$
   • C$3025$
   • D$2310$

   (c)

   Find the value of $\sum_{r=1}^{12}r^3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $S_n=\sum_{r=1}^{n}\left(r^2+r\right)$.

   (a)

   Find the value of $S_6$.

   [1 mark]

   • A$91$
   • B$21$
   • C$112$
   • D$546$

   (b)

   Which expression equals $S_n$?

   [1 mark]

   • A$\frac{n(n+1)(n+2)}{6}$
   • B$\frac{n(n+1)(2n+1)}{3}$
   • C$\frac{n(n+1)(n+2)}{2}$
   • D$\frac{n(n+1)(n+2)}{3}$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(n)=\sum_{r=1}^{n}r\left(r^2+2\right)$.

   (a)

   Show that $f(n)=\frac14n(n+1)\left(n^2+n+4\right)$.

   [3 marks]

   (b)

   Hence find the value of $\sum_{r=10}^{20}r\left(r^2+2\right)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The $r$th term of a sequence is $u_r=(2r-1)^2$.

   (a)

   Show that $\sum_{r=1}^{n}u_r=\frac13n\left(4n^2-1\right)$.

   [6 marks]

   (b)

   (i) Hence find the value of $51^2+53^2+55^2+\cdots+99^2$.
   (ii) Find, in terms of $n$, $\sum_{r=1}^{n}(2r-1)(2r)$, giving your answer in a fully factorised form.

   [6 marks]

   Total for question 4: 12 marks

End of questions