A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by a_{n+1} = 5a_n - 3, \quad n \geq 1 Given that $a_2 = 7,$ (a) find the value of $a_1$ (b) Find the value of $\sum_{r=1}^{4} a_r$ - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 1

Now we need to find $a_3$ and $a_4$ to compute the sum.

Using the recurrence relation again:

For $n = 2$:

$a_3 = 5a_2 - 3 = 5(7) - 3 = 35 - 3 = 32$

For $n = 3$:

$a_4 = 5a_3 - 3 = 5(32) - 3 = 160 - 3 = 157$

Now we have:

• $a_1 = 2$
• $a_2 = 7$
• $a_3 = 32$
• $a_4 = 157$

Now we can find the sum:

$\sum_{r=1}^{4} a_r = a_1 + a_2 + a_3 + a_4$

$= 2 + 7 + 32 + 157 = 198$