Geometric sequences and seriesEdexcel International A Level Maths: Subtopic test

1. 1
   A geometric series has first term $6$ and common ratio $\frac12$.

   (a)

   Find the 5th term.

   [1 mark]

   • A$\frac{3}{16}$
   • B$\frac32$
   • C$\frac38$
   • D$3$

   (b)

   Find the sum to infinity.

   [1 mark]

   • A$4$
   • B$6$
   • C$8$
   • D$12$

   (c)

   Find the exact sum of the first $8$ terms.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A geometric series has first term $5$ and common ratio $1.2$. Its $n$th term is $u_n$.

   (a)

   Find the value of $u_{10}$, to 3 significant figures.

   [1 mark]

   • A$25.8$
   • B$31.0$
   • C$60.0$
   • D$54.0$

   (b)

   Find the smallest value of $n$ for which $u_n>1000$.

   [1 mark]

   • A$30$
   • B$31$
   • C$29$
   • D$32$

   (c)

   Find the sum of the first $15$ terms, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A geometric series with positive common ratio has second term $24$ and fourth term $6$. The sum of the first $n$ terms is $S_n$.

   (a)

   Find the first term and the common ratio.

   [3 marks]

   (b)

   Find the sum to infinity, and the smallest value of $n$ for which the sum to infinity exceeds $S_n$ by less than $0.001$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A geometric series has first term $a$ and common ratio $r$, where $r\ne1$. The sum of the first $n$ terms is $S_n$.

   (a)

   Prove that $S_n=\frac{a(1-r^n)}{1-r}$.

   [6 marks]

   (b)

   Given that the second term is $12$ and the sum to infinity is $64$, find the two possible values of $r$ and the corresponding values of $a$.

   [6 marks]

   Total for question 4: 12 marks

End of questions