A sequence of numbers $a_1, a_2, a_3, \ldots$ is defined by $$a_{n+1} = 5a_n - 3, \ n > 1$$ 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

We first need to determine the first four terms of the sequence:

• We have already found $a_1 = 2$ and $a_2 = 7$.
• From earlier, we calculated $a_3 = 32$. Now, let's find $a_4$ by substituting $n=3$:

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

Now we have:

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

We can now compute the sum:

$\sum_{r=1}^{4} a_r = a_1 + a_2 + a_3 + a_4 = 2 + 7 + 32 + 157$
Calculating the sum step by step:

• $2 + 7 = 9$
• $9 + 32 = 41$
• $41 + 157 = 198$

Thus, the value of the sum is 198.