Summation of seriesAQA A-Level Further Maths: Subtopic test

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

   (a)

   What is the value of $S_4$?

   [1 mark]

   • A$26$
   • B$30$
   • C$20$
   • D$50$

   (b)

   Which expression is equal to $S_n$?

   [1 mark]

   • A$\frac{n(n+1)(2n+7)}{6}$
   • B$\frac{n(n+1)(2n+5)}{6}$
   • C$\frac{n(n+1)(2n+1)}{6}+2n$
   • D$\frac{n(n+1)(n+2)}{3}$

   (c)

   Find the value of $\sum_{r=11}^{20}r(r+2)$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   What is the value of $U_3$?

   [1 mark]

   • A$30$
   • B$18$
   • C$54$
   • D$36$

   (b)

   What is the value of $U_{10}$?

   [1 mark]

   • A$2970$
   • B$3190$
   • C$2860$
   • D$3025$

   (c)

   Show that $U_n=\frac14n(n+1)(n-2)(n+3)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(r)=r(r+1)(r+2)$.

   (a)

   Show that $f(r)-f(r-1)=3r(r+1)$.

   [3 marks]

   (b)

   Hence use the method of differences to show that $\sum_{r=1}^{n}r(r+1)=\frac{n(n+1)(n+2)}{3}$.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   (i) Show that $\frac1r-\frac1{r+2}=\frac2{r(r+2)}$.
   (ii) Hence show that $S_n=\frac34-\frac{2n+3}{2(n+1)(n+2)}$.

   [6 marks]

   (b)

   (i) Find the exact value of $\sum_{r=10}^{20}\frac1{r(r+2)}$.
   (ii) Find the smallest value of $n$ for which $\frac34-S_n<0.001$.

   [6 marks]

   Total for question 4: 12 marks

End of questions