The 8th term of an arithmetic sequence is 36. If the common difference is 2, what is the first term in the sequence?

We are given that the 8th term of an arithmetic sequence is 36, and the common difference is 2, and we are asked to find the first term:

In an arithmetic sequence, each term can be found by adding the common difference to the previous term. Thus we start with a(1), then a(2)=a(1)+d. Now a(3)=a(2)+d=(a(1)+d)+d=a(1)+2d.

The general form is `a_n=a_1+(n-1)d`

where n is the number of terms and d is the common difference.

Here we have a(8)=36; substituting the known values we get:

36=a(1)+7(2) ==> a(1)= 22

The first term is 22.

We can easily check this: 22,24,26,28,30,32,34,36.

Another way to think of this is to recognize an arithmetic sequnce as a set of points on a line. We are given the constant rate of change, 2, which is the slope of the line. We are also give a point on the line which is (8,36).

So y-36=2(x-8) or y=2x+20. So when x=0 we have y=20; when x=1 (the first term in our sequence) we have y=22.