Question 1158007: Hi
The sum of 3 terms of an arithmetic progression is 42 and the product of the
First and third term is 52.
Find the 3 terms.
Thanks
Found 3 solutions by josgarithmetic, ikleyn, MathTherapy:
Answer by josgarithmetic(39616)   (Show Source):
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
This problem is nice. But it becomes nice only in hands of those who knows how to solve it and
knows how to present the solution to others in a way they see its beaty.
It is not the way @josgarithmetic tries to do it: he simply does not know the right way.
By following his way, you will NEVER feel its beauty.
Let "a" be the middle term of the progression, and let "d" be its common difference.
Then the first term of the AP is  = d-a, and its third term is  = d+a.
Since the sum ot the three terms is equal to 42, you have this equation
(a-d) + a + (a+d) = 42,
or 3a = 42, which immediately implies a = 42/3 = 14.
Thus we just found the middle terms, very quickly, easy and practically MENTALLY.
Next, regarding the product of the terms, we have this equation
(a-d)*(a+d) = 52,
or
a^2 - d^2 = 52.
Substitute here a = 14, the value found couple of lines above, and you will get
d^2 = a^2 - 52 = 14^2 - 52 = 144;
therefore
d = +/-  = +/- 12.
At this stage, the problem is just solved.
ANSWER. For the three terms, there are two possibilities.
1) The progression is 14-12 = 2, 14 and 14+12 = 26 ( with d = 12 ), or
2) The progression is 14+12 = 26, 14 and 14-12 = 2 ( with d = -12 ).
So, the second progression is simply the reversed first progression.
Solved.
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It is how this problem is designed, is intended and is expected to be solved.
Answer by MathTherapy(10551)  (Show Source):
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