SOLUTION: The first and second terms of an arithmetic sequence are a and b respectively. If the nth term is c, express n in terms of a b and c and hence find the sum of these n terms.

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Question 1124987: The first and second terms of an arithmetic sequence are a and b respectively. If the nth term is c, express n in terms of a b and c and hence find the sum of these n terms.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The first term is a; the second is b; so the common difference is (b-a).

The n-th term is then the first term plus the common difference (n-1) times:
c+=+a+%2B+%28n-1%29%28b-a%29
c+=+a+%2B+nb+-+na+-+b+%2B+a
c+=+2a-b%2Bnb-na
b%2Bc-2a+=+n%28b-a%29
n+=+%28b%2Bc-2a%29%2F%28b-a%29 <-- answer to the first question

The sum of the series is (number of terms) times (average of first and last terms):

S+=+n%28%28a%2Bc%29%2F2%29
S+=+%28%28b%2Bc-2a%29%2F%28b-a%29%29%28%28a%2B%282a-b%2Bnb-na%29%29%2F2%29
S+=+%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29 <-- answer to the second question

Let's check these formulas with an example -- just in case we made an error in our algebra....

a = 3; b = 5; 9 terms.

The common difference is b-a = 2; the 9th term, c, is 3+8(2) = 19.

We know the number of terms is 9; our formula says the number of terms should be

%28b%2Bc-2a%29%2F%28b-a%29
%285%2B19-2%283%29%29%2F%285-3%29+=+18%2F2+=+9

Our formula gives the right result for this example.

The sum of the series is 9%28%283%2B19%29%2F2%29+=+9%2A11+=+99.

Our formula says the sum is supposed to be

%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29
%28%285%2B19-2%283%29%29%283%283%29-5%2B9%285%29-9%283%29%29%29%2F%282%285-3%29%29
%28%2824-6%29%289-5%2B45-27%29%29%2F4+=+%2818%2A22%29%2F4+=+9%2A11+=+99

Our formula for the sum also gives the right result for this example.