Geometric seriesAQA A-Level Maths: Subtopic test

1. 1
   A geometric sequence has first term $2$ and common ratio $3$.

   (a)

   Find the $5$th term.

   [1 mark]

   • A$486$
   • B$162$
   • C$54$
   • D$30$

   (b)

   Find the sum of the first $6$ terms.

   [1 mark]

   • A$-728$
   • B$242$
   • C$1456$
   • D$728$

   (c)

   Find the least value of $n$ for which the sum of the first $n$ terms exceeds $10^6$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A geometric series has first term $40$ and common ratio $0.6$.

   (a)

   Find the sum to infinity.

   [1 mark]

   • A$100$
   • B$25$
   • C$60$
   • D$\frac{200}{3}$

   (b)

   Find the sum of the first $10$ terms, correct to $3$ significant figures.

   [1 mark]

   • A$99.0$
   • B$100$
   • C$99.4$
   • D$66.3$

   (c)

   Find the least value of $n$ for which $S_\infty-S_n<0.001$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A geometric series has second term $12$ and fifth term $1.5$.

   (a)

   Find the common ratio and the first term.

   [3 marks]

   (b)

   Find the sum to infinity and the sum of the terms from the $5$th term onwards.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A geometric series has first term $a$ and common ratio $r$, where $|r|<1$.

   (a)

   The sum to infinity is $4$ times the second term.
   (i) Show that $(2r-1)^2=0$ and write down the value of $r$.
   (ii) Given that $a=6$, find the least value of $n$ for which the sum of the first $n$ terms exceeds $11.99$.

   [6 marks]

   (b)

   A different geometric series is $3+3(2x-1)+3(2x-1)^2+\dots$
   (i) Find the values of $x$ for which the series converges, using modulus notation in your working.
   (ii) Find $S_\infty$ in terms of $x$.
   (iii) Given that $S_\infty=5$, find $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions