Given $u_1 = 1$, determine which one of the formulae below defines an increasing sequence for $n \geq 1$ - AQA - A-Level Maths Pure - Question 3 - 2019 - Paper 3

To identify which formula results in an increasing sequence, we analyze each option starting with the known initial condition $u_1 = 1$:

1. $u_{n+1} = 1 + \frac{1}{u_n}$:

   • Calculate the first few terms:
     • $u_2 = 1 + \frac{1}{1} = 2$,
     • $u_3 = 1 + \frac{1}{2} = 1.5$.
   • This does not yield an increasing sequence.

2. $u_{n+1} = 2 - 0.9^{n-1}$:

   • Calculate:
     • $u_2 = 2 - 0.9^{1} = 1.1$,
     • $u_3 = 2 - 0.9^{2} = 1.81$.
   • The sequence is increasing since each term is greater than the previous when calculated further.

3. $u_{n+1} = -1 + 0.5u_n$:

   • Calculate:
     • $u_2 = -1 + 0.5(1) = -0.5$.
   • This formula does not yield an increasing sequence.

4. $u_n = 0.9^{n-1}$:

   • It defines a decreasing sequence as powers of a number less than 1 lead to diminishing values.

Thus, the formula that defines an increasing sequence is:

$u_{n+1} = 2 - 0.9^{n-1}$.