SOLUTION: The Sum of first 20th terms is 200 and Sum of next 10 terms is 400.What is the 40th term?

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Question 1092302: The Sum of first 20th terms is 200 and Sum of next 10 terms is 400.What is the 40th term?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

You don't say so, but I will assume this is an arithmetic sequence....

In any arithmetic sequence, the sum of n terms is n times the average of the first and last terms.

The sum of the first 20 terms is 20 times the average of the first and 20th terms. If the first term is a and the common difference is d, then the 20th term is a+19d. Since your problem says the sum of the first 20 terms is 200,
20%28a%2B%28a%2B19d%29%29%2F2+=+200
10%282a%2B19d%29+=+200
2a%2B19d+=+20 (1)

Also in your problem the sum of the next 10 terms (terms 21 through 30) is 400. The 21st term is a+20d; the 30th is a+29d. The sum of those 10 terms is
10%28%28a%2B20d%29%2B%28a%2B29d%29%29%2F2+=+400
2a%2B49d+=+80 (2)

Subtract equation (1) from equation (2) and solve for d; then plug that value for d in either (1) or (2) to solve for a.

30d+=+60
d+=+2
2a%2B19%282%29+=+20
2a%2B38+=+20
2a+=+-18
a+=+-9

We are to find the 40th term, which is a+39d:
-9%2B40%282%29+=+-9%2B80+=+71

The 40th term is 71.

Oops! I said the 40th term is a+39d, but then I calculated a+40d.

The 40th term is a+39d:
-9%2B39%282%29+=+-9%2B78+=+69

So the 40th term is 69, which is one of your answer choices.