Geometric sequences and seriesEdexcel A-Level Maths: Subtopic test

1. 1
   A geometric sequence has first term $64$ and common ratio $\frac34$.

   (a)

   Find the $5$th term.

   [1 mark]

   • A$\frac{243}{16}$
   • B$67$
   • C$\frac{81}{256}$
   • D$\frac{81}{4}$

   (b)

   Find the sum to infinity of the corresponding series.

   [1 mark]

   • A$\frac{256}{7}$
   • B$\frac{256}{3}$
   • C$256$
   • DThe sum to infinity does not exist.

   (c)

   Find the least value of $n$ for which the $n$th term is less than $1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A geometric series has second term $12$ and fifth term $324$.

   (a)

   Find the common ratio.

   [1 mark]

   • A$4$
   • B$3$
   • C$9$
   • D$27$

   (b)

   Find the first term.

   [1 mark]

   • A$4$
   • B$36$
   • C$12$
   • D$\frac43$

   (c)

   Find the sum of the first $8$ terms.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A geometric series has first term $a$ and common ratio $r$, where $r\ne1$.

   (a)

   Prove that the sum of the first $n$ terms is $S_n=\frac{a(1-r^n)}{1-r}$.

   [3 marks]

   (b)

   The series converges, its sum to infinity is $48$ and its second term is $9$. Find the two possible values of $r$ and the corresponding values of $a$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A geometric sequence has first term $800$ and common ratio $0.9$.

   (a)

   (i) Show that $S_n=8000(1-0.9^n)$.
   (ii) Find the sum to infinity.
   (iii) Find the smallest value of $n$ for which $S_n>7500$.

   [6 marks]

   (b)

   (i) Show that the sum of all the terms from the $10$th term onwards is $8000\times0.9^{9}$.
   (ii) Find the least value of $m$ for which the sum of the terms from the $m$th term onwards is less than $100$.

   [6 marks]

   Total for question 4: 12 marks

End of questions