A sequence $a_n, a_{n+1}, a_{n+2},\, \ldots$ is defined by $a_n = 4a_{n-1} - 3, \quad n > 1$ $a_1 = k,$ where $k$ is a positive integer. (a) Write down an expression for $a_2$ in terms of $k.$ Given that $\sum_{n=1}^3 a_n = 66$ (b) find the value of $k.$

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A sequence $a_n, a_{n+1}, a_{n+2},\, \ldots$ is defined by $a_n = 4a_{n-1} - 3, \quad n > 1$ $a_1 = k,$ where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 2

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$A-sequence-$a_n,-a_{n+1},-a_{n+2},\,-\ldots$-is-defined-by---$a_n-=-4a_{n-1}---3,-\quad-n->-1$---$a_1-=-k,$-where-$k$-is-a-positive-integer-Edexcel-A-Level Maths Pure-Question 5-2014-Paper 2.png$

A sequence $a_n, a_{n+1}, a_{n+2},\, \ldots$ is defined by $a_n = 4a_{n-1} - 3, \quad n > 1$ $a_1 = k,$ where $k$ is a positive integer. (a) Write down an exp... show full transcript

Worked Solution & Example Answer:A sequence $a_n, a_{n+1}, a_{n+2},\, \ldots$ is defined by $a_n = 4a_{n-1} - 3, \quad n > 1$ $a_1 = k,$ where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 2

Step 1

Write down an expression for $a_2$ in terms of $k.$

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Answer

To find the expression for $a_2$ , we use the given formula:

$a_2 = 4a_1 - 3$
Substituting $a_1 = k$ , we have:

$a_2 = 4k - 3.$

Step 2

find the value of $k.$

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Answer

We know that:

$\sum_{n=1}^3 a_n = a_1 + a_2 + a_3 = 66.$
Replacing $a_1$ and $a_2$ with their expressions:

$a_3 = 4a_2 - 3 = 4(4k - 3) - 3 = 16k - 12 - 3 = 16k - 15.$
Now substituting these into the summation:

$k + (4k - 3) + (16k - 15) = 66.$
Combining like terms:

$21k - 18 = 66.$
Solving for $k$ :

$21k = 84 \implies k = 4.$
Thus, the value of $k$ is $4$ .

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