Question

What is the sum of a 12–term arithmetic sequence where the last term is 13 and the common difference is –10?

Answer 1

ok, so here you are given \(n=12\), \(a_{12}=13\) and \(d=-10\)

Answer 2

now, do you recall the formula for calculating the n'th term of a sequence?

Answer 3

yes.

Answer 4

you can use that formula to work out \(a_1\)

Answer 5

once you have that, then you can work out the sum \(S_{12}\)

Answer 6

Is the first term 3?

Answer 7

no

Answer 8

What formula do I use to find the first term?

Answer 9

ok, think about how would solve this problem.
you are asked to find the sum of a 12 term series, and you of two formulas for the sum:\[S_n=\frac{n}{2}(a_1+a_n)\tag{1}\]\[S_n=\frac{n}{2}(2a_1+(n-1)d)\tag{2}\]you are given the values of \(n\), \(a_n\) and \(d\).

so which of these two formulas would you pick to get the sum?

Answer 10

can you see that in both formulas you need to know the value of \(a_1\)?

Answer 11

yes, so how would you find the first term?

Answer 12

good - so you are thinking now :)

recall the formula for the n'th term:\[a_n=a_1+(n-1)d\tag{3}\]in this formula you know everything except \(a_1\) - which is exactly the value you need in order to use either formula (1) or (2) above.

so you can first re-arrange formula (3) to get an expression for \(a_1\) as follows:\[a_1=a_n-(n-1)d\tag{4}\]so now you can use this formula to work out \(a_1\), and then use either formula (1) or (2) to work out the sum.

Answer 13

133?

Answer 14

what is 133?

Answer 15

first term?

Answer 16

can you please show your working?

Answer 17

13 - 12(-10)

Answer 18

look carefully at the formula you are using

Answer 19

\(d\) is multiplied by \((n-1)\) not by \(n\)

Answer 20

so it would be 11?

Answer 21

13 - (11)(-10)

Answer 22

correct

Answer 23

123?

Answer 24

yes - that is the first term

Answer 25

final answer = 816?

Answer 26

perfect! well done again! :)