Question

What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term
is 13 ?

Answer 1

Can you tell me the formula for the nth term of an arithmetic sequence?

Answer 2

Its an act question. Thats all it gives me

Answer 3

What do you think the answer is?

Answer 4

@debins33, I know that's all they give you but in order to solve it you're going to need to know a few formulas - the first of which being the formula for the nth term of an arithmetic sequence. Look it up and tell me when you find it.

Answer 5

a+(n-1)d

Answer 6

All of them seem to involve knowing the first term

Answer 7

The ACT says the answer is 14.5

Answer 8

That's it! Great job. So now we look at the information in the question:

"What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is 13 ?"

From this, we know that:
\[a_6=8=a+(6-1)d=a+5d\]\[a_{10}=13=a+(10-1)d=a+9d\]
As you said, we need to find the first term. To do this, we can just solve these two equations simultaneously:
\[8=a+5d\]\[13=a+9d\]
Can you solve these for a and d?

Answer 9

Oh okay I see how that works. I'm a bit rusty with my algebra would I put it in terms of d or a to solve?

Answer 10

Great, glad you understand. There's two ways you could solve those: rearranging and just subtracting.

Rearranging:
\[8=a+5d, a=8-5d\]\[13=a+9d, a=13-9d\]\[8-5d=13-9d\]\[4d=5\]\[d=1.25\]\[a=8-5*1.25=1.75\]

Subtracting:
\[8=a+5d, 13=a+9d\]
\[13-8=9d-5d\]\[5=4d\]\[d=1.25\]\[a=8-5*1.25=1.75\]

Answer 11

Now we have the first term and the common difference, that's all we need to know! Can you tell me the equation for the sum of n terms in an arithmetic sequence?

Answer 12

(n/2)(a1+an)

Answer 13

Thank you!

Answer 14

Yep, so we can work out a4 using the a and d we just worked out, then it's simply a case of subbing those numbers into that equation :)