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

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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...

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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.

 

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