1. 1
   A geometric sequence has first term $u_1 = 24$ and common ratio $r = -\frac{1}{2}$.

   (a)

   Find $u_4$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$6$
   • D$1.5$

   (b)

   Find the sum to infinity of the sequence.

   [1 mark]

   • A$48$
   • B$16$
   • C$-16$
   • D$12$

   (c)

   Find the sum to infinity of the odd-numbered terms, $u_1 + u_3 + u_5 + \cdots$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the infinite geometric series $1 + (2x-1) + (2x-1)^2 + (2x-1)^3 + \cdots$, where $x\in\mathbb{R}$.

   (a)

   Find the set of values of $x$ for which the series converges.

   [1 mark]

   • A$-1<x<1$
   • B$x<1$
   • C$0<x<1$
   • D$-\frac12<x<\frac12$

   (b)

   Find the sum to infinity of the series when $x = 0.3$.

   [1 mark]

   • A$\frac{5}{3}$
   • B$\frac{7}{5}$
   • C$\frac{10}{7}$
   • D$\frac{5}{7}$

   (c)

   Given that the sum to infinity of the series is $\frac43$, find the value of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is dropped from a height of 3 m onto a hard floor. After each bounce it rises vertically to 60% of the height from which it last fell. Assume the ball continues to bounce indefinitely.

   (a)

   Show that the total vertical distance travelled by the ball is 12 m.

   [3 marks]

   (b)

   A second ball is also dropped from 3 m, but after each bounce it rises to a fraction $p$ of the height from which it last fell, where $0<p<1$. The total vertical distance it travels is 7 m. Find $p$, and justify that your value is consistent with the model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An infinite geometric series has first term $a$ and common ratio $r$. The sum to infinity of the series is 45 and the sum of its first two terms is 40.

   (a)

   Show that $1 - r^2 = \frac{8}{9}$, and hence find the two possible pairs of values of $a$ and $r$.

   [6 marks]

   (b)

   It is now given that every term of the series is positive.
   (i) Write down the values of $a$ and $r$.
   (ii) Find the least number of terms that must be added for the sum to exceed 44.
   (iii) Explain why the sum of a finite number of terms can never exceed 45.

   [6 marks]

   Total for question 4: 12 marks

End of questions