A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant. (a) Find an expression for $a_2$ in terms of $k$. Given that $\sum_{i=1}^3 a_i = 2$, (b) find the two possible values of $k$.

A-Level Edexcel Maths Pure Simultaneous Equations: A sequence $a_n$, $a_{n+1}$, $a

A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Question 6

$A-sequence-$a_n$,-$a_{n+1}$,-$a_{n+2},-...$-is-defined-by--$a_1-=-4$-$a_{n+1}-=-k(a_n-+-2)$-for-$n--geq-1$--where-$k$-is-a-constant-Edexcel-A-Level Maths Pure-Question 6-2013-Paper 1.png$

A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant. (a) Find an expression for $a_2... show full transcript

Worked Solution & Example Answer:A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Step 1

Find an expression for $a_2$ in terms of $k$.

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Answer

To find $a_2$ , we substitute $n=1$ into the recurrence relation:

$a_{2} = k(a_{1} + 2)$

Substituting $a_{1} = 4$ , we have:

$a_{2} = k(4 + 2) = 6k.$

Step 2

find the two possible values of $k$.

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Answer

Given that: $\sum_{i=1}^3 a_i = a_1 + a_2 + a_3 = 2,$ we first find expressions for $a_3$ using the recurrence relation:

$a_3 = k(a_2 + 2) = k(6k + 2) = 6k^2 + 2k.$

Now substituting for $a_1$ , $a_2$ , and $a_3$ :

$4 + 6k + (6k^2 + 2k) = 2.$

This simplifies to:

ightarrow 6k^2 + 8k + 2 = 0.$$ Now we solve this quadratic equation using the quadratic formula: $$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 6 \cdot 2}}{2 \cdot 6}$$ Calculating the discriminant: $$b^2 - 4ac = 64 - 48 = 16,$$ so: $$k = \frac{-8 \pm 4}{12}.$$ Thus we find two values: 1. $$k = \frac{-4}{12} = -\frac{1}{3},$$ 2. $$k = \frac{-12}{12} = -1.$$

A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Worked Solution & Example Answer:A sequence $a_n$, $a_{n+1}$, $a_{n+2}, ...$ is defined by $a_1 = 4$ $a_{n+1} = k(a_n + 2)$ for $n geq 1$ where $k$ is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Find an expression for $a_2$ in terms of $k$.

find the two possible values of $k$.

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