OCR MEI C2 (Core Mathematics 2)

1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).

2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 ,
u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

1. Find the third term of this sequence and state what type of sequence it is.
2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.

3 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 48th term of this sequence.
2. Find the sum of the first 48 terms of this sequence.

4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

6

1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
3. Sketch the curve with equation \(y = 2 ^ { x }\).