An arithmetic sequence has first term a and common difference d. The sum of the first 10 terms of the sequence is 162. (a) Show that 10a + 45d = 162 Given also that the sixth term of the sequence is 17, (b) write down a second equation in a and d, (c) find the value of a and the value of d.

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An arithmetic sequence has first term a and common difference d - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 2

Question 8

An arithmetic sequence has first term a and common difference d. The sum of the first 10 terms of the sequence is 162. (a) Show that 10a + 45d = 162 Given also tha... show full transcript

Worked Solution & Example Answer:An arithmetic sequence has first term a and common difference d - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 2

Step 1

Show that 10a + 45d = 162

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Answer

To find the sum of the first 10 terms of an arithmetic sequence, we can use the formula:

$S_n = \frac{n}{2} [2a + (n - 1)d]$

For the first 10 terms, we have:

$S_{10} = \frac{10}{2} [2a + (10 - 1)d]$

This simplifies to:

$S_{10} = 5[2a + 9d]$

Setting this equal to the sum, we get:

$5[2a + 9d] = 162$

Dividing both sides by 5:

$2a + 9d = 32.4$

To express this in terms of 10a + 45d, we can multiply the entire equation by 5:

$10a + 45d = 162$

Thus, we have shown that the equation is valid.

Step 2

write down a second equation in a and d

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Answer

The sixth term of an arithmetic sequence can be calculated using the formula:

$u_n = a + (n - 1)d$

For the sixth term (n = 6), we have:

$u_6 = a + 5d$

Given that the sixth term is 17, we can write:

$a + 5d = 17$

Step 3

find the value of a and the value of d

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Answer

We now have two equations:

$10a + 45d = 162$
$a + 5d = 17$

From the second equation, we can express a in terms of d:

$a = 17 - 5d$

Now we substitute this expression for a into the first equation:

$10(17 - 5d) + 45d = 162$

Expanding this gives:

$170 - 50d + 45d = 162$

Combining like terms results in:

$170 - 5d = 162$

Subtracting 170 from both sides:

$-5d = -8$

Dividing by -5:

$d = 1.6$

Now substituting the value of d back into the expression for a:

$a = 17 - 5(1.6)$

Calculating this yields:

$a = 17 - 8 = 9$

Thus, the values are:

$a = 9$
$d = 1.6$

An arithmetic sequence has first term a and common difference d - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 2

Worked Solution & Example Answer:An arithmetic sequence has first term a and common difference d - Edexcel - A-Level Maths Pure - Question 8 - 2011 - Paper 2

Show that 10a + 45d = 162

write down a second equation in a and d

find the value of a and the value of d

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