October 2021 Paper 1 Q4
4
(a) The first four terms of a sequence are 2, 3, 0, 3 and the subsequent terms are given by \(a_{k+4} = a_k\).
(i) State what type of sequence this is. [1]
(ii) Find \(\displaystyle\sum_{k=1}^{200} a_k\). [1]
(b) A different sequence is given by \(u_n = b^n\) where \(b\) is a constant and \(n \geqslant 1\).
(i) State the set of values of \(b\) for which this is a divergent sequence. [2]
(ii) In the case where \(b = \frac{1}{3}\), find the sum of all the terms in the sequence. [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) Sequence is periodic [with period 4] | B1 | 1.2 |
| [1] | ||
| (ii) Total of 200 terms is \(50 \times (2 + 3 + 0 + 3) = 400\) | B1 | 1.1b |
| [1] |
Notes
(i) B1: Do not allow repeating, recurring etc.
The sequence can also be described as oscillating
(ii) B1: cao
| Scheme | Marks | AO |
|---|---|---|
| (i) Sequence divergent for Either \(b > 1\) | B1 | 1.1b |
| or \(b \leqslant -1\) | B1 | 1.1b |
| [2] | ||
| (ii) Infinite sum of geometric series with \(a = \frac{1}{3},\ r = \frac{1}{3}\) | M1 | 1.1b |
| \(S = \dfrac{a}{1-r} = \dfrac{\frac{1}{3}}{1-\frac{1}{3}} = \dfrac{1}{2}\) | A1 | 1.1b |
| [2] |
Notes
(i) B1: Allow for one correct inequality
B1: Must have “or” or the union of sets
Condone \(b < -1\) or \(|b| > 1\)
Note for \(b = 1\), the sequence is convergent, but the corresponding series is divergent
(ii) M1: Using the sum of geometric series with \(r = \frac{1}{3}\)
A1: www