Consecutive terms of a sequence are related by $$u_{n+1} = 3 - (u_n)^2$$ In the case that $u_1 = 2$ 7 (a) (i) Find $u_3$ 7 (a) (ii) Find $u_{50}$ 7 (b) State a different value for $u_1$ which gives the same value for $u_{50}$ as found in part (a)(ii). - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 1

From the recurrence relation, we see that:

• $u_1 = 2$, $u_2 = -1$, $u_3 = 2$, $u_4 = -1$, and so on.

We observe that the terms oscillate between $2$ and $-1$. Thus, for even indices, we have:

• $u_{2n} = -1$ (for $n eq 0$)

Since $50$ is even: $u_{50} = -1$

To find another value for $u_1$ that gives $u_{50} = -1$, we can consider:

• $u_1 = -2$ leads to:
  1. Calculate $u_2$: $u_2 = 3 - (-2)^2 = 3 - 4 = -1$
  2. Then, we see:
     • $u_3 = 2$, $u_4 = -1$, and the sequence continues oscillating.

Therefore, another possible value for $u_1$ is $-2$.