A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer. (a) Write down an expression for $a_2$ in terms of $k$. (b) Show that $a_3 = 25k + 18.$ (c) (i) Find $ \sum_{n=1}^{4} a_n $ in terms of $k$, in its simplest form. (ii) Show that $ \sum_{n=1}^{4} a_n $ is divisible by 6.

A-Level Edexcel Maths Pure Quadratics: A sequence $a_1, a_2, a

A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 1

Question 7

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A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer. (a) Write down an expression for $a_2$ in t... show full transcript

Worked Solution & Example Answer:A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 1

Step 1

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

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Answer

$a_2 = 5k + 3$

Step 2

Show that $a_3 = 25k + 18$.

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Answer

To find $a_3$ , we use the expression for $a_2$ :

$a_3 = 5a_2 + 3 = 5(5k + 3) + 3.$
Expanding this, we have:

$a_3 = 25k + 15 + 3 = 25k + 18.$

Step 3

Find $ \sum_{n=1}^{4} a_n $ in terms of $k$, in its simplest form.

96%

101 rated

Answer

We first find each term:

$a_1 = k$
$a_2 = 5k + 3$
$a_3 = 25k + 18$
For $a_4$ : $a_4 = 5a_3 + 3 = 5(25k + 18) + 3 = 125k + 90 + 3 = 125k + 93.$
Now we can compute the sum:

$\sum_{n=1}^{4} a_n = k + (5k + 3) + (25k + 18) + (125k + 93) = (1 + 5 + 25 + 125)k + (3 + 18 + 93) = 156k + 114.$

Step 4

Show that $ \sum_{n=1}^{4} a_n $ is divisible by 6.

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120 rated

Answer

We have $\sum_{n=1}^{4} a_n = 156k + 114.$
Factoring this: $156k + 114 = 6(26k + 19).$
Thus, ( \sum_{n=1}^{4} a_n ) is divisible by 6 since it can be expressed as 6 times an integer.

A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 1

Worked Solution & Example Answer:A sequence $a_1, a_2, a_3, ...$ is defined by a_1 = k, a_{n+1} = 5a_n + 3, n \geq 1, where $k$ is a positive integer - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 1

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

Show that $a_3 = 25k + 18$.

Find \( \sum_{n=1}^{4} a_n \) in terms of $k$, in its simplest form.

Show that \( \sum_{n=1}^{4} a_n \) is divisible by 6.

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