SOLUTION: The sum of the first twenty terms in an arithmetic progression is 840 and the common difference is 4.Find (a)the first term (b)the sum of the first ten terms

Question 1094750: The sum of the first twenty terms in an arithmetic progression is 840 and the common difference is 4.Find
(a)the first term
(b)the sum of the first ten terms
Found 3 solutions by josgarithmetic, MathTherapy, greenestamps:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
The sum of the first twenty terms in an arithmetic progression is 840 and the common difference is 4.Find
(a)the first term
(b)the sum of the first ten terms

The 1st term IS NOT 12, as "you-know-who" states!
Try it yourself! You may get the correct answer, and not be led astray!
You should get a 1st term of 4. Hope you can do it!!
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

A key concept for working with arithmetic progressions is that we can group the terms in pairs, with each pair having the same sum.

In your example, where the sum of the first 20 terms is 840, we know those 20 terms are 10 pairs, so the sum of each pair is 840/10 = 84.

One of the pairs is the first and last (20th) terms. The 20th term is the first term, plus the common difference (which is 4) 19 times; and the sum of the first and 20th terms is 84. So
--> a = 4.

That's the answer to part (a).

Part (b): In finding the sum of the first 10 terms, we will have 5 pairs with the same sum; one of those pairs is the 1st and 10th terms. The 10th term is the first term 4, plus the common difference (4) 9 times; 4+9(4) = 40. So the sum of each of the 5 pairs is 4+40 = 44; since there are 5 pairs, the sum of the first 10 terms is 44*5 = 220.
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