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, we use the formula:

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

For this sequence, with n = 10:

$S_{10} = \frac{10}{2} \left[2a + (10-1)d\right] = 5[2a + 9d]$

Setting this equal to 162, we get:

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

Dividing both sides by 5 gives:

$2a + 9d = 32.4$

Multiplying by 5 to eliminate the fraction, we can expand:

ightarrow 10a + 45d = 162$$ Thus, we have shown that the equation holds true.

Step 2

write down a second equation in a and d

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Answer

The sixth term of the sequence can be expressed as:

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

For n = 6, this becomes:

$u_6 = a + 5d$

Given that this term equals 17, the equation becomes:

$a + 5d = 17$ .

Step 3

find the value of a and the value of d

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Answer

Now, we have a system of two equations:

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

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

$a = 17 - 5d$

Substituting this into the first equation gives:

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

Expanding and simplifying:

ightarrow 170 - 5d = 162$$ Rearranging leads to: $$-5d = -8 ightarrow d = 1.6$$ Now substituting d back into the equation for a: $$a = 17 - 5(1.6)$$ Calculating gives: $$a = 17 - 8 = 9$$ Thus, the values are: **a = 9** and **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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