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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 $a_1$ - Edexcel - A-Level Maths Pure - Question 7 - 2014 - Paper 1

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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 th... show full transcript

Worked Solution & Example Answer: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 $a_1$ - Edexcel - A-Level Maths Pure - Question 7 - 2014 - Paper 1

Step 1

(a) find the value of $a_1$

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Answer

To find the value of $a_1$ , we start with the recurrence relation:

$a_{n+1} = 5a_n - 3$

Given that $a_2 = 7$ , we substitute $n = 2$ :

$a_{2} = 5a_{1} - 3$

This gives us:

$7 = 5a_{1} - 3$

Adding 3 to both sides results in:

$10 = 5a_{1}$

Dividing both sides by 5 yields:

$a_1 = 2$

Step 2

(b) Find the value of $\sum_{r=1}^{4} a_r$

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Answer

First, we need to find $a_3$ and $a_4$ using the recurrence relation:

For $n = 2$ :

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

Next, for $n = 3$ :

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

Now we can find the sum:

$\sum_{r=1}^{4} a_r = a_1 + a_2 + a_3 + a_4 = 2 + 7 + 32 + 157 = 198$

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