SOLUTION: The sum of the first five terms of an AP is 10, the sum of their squares is 380. Find the first term and common difference?

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Question 979908: The sum of the first five terms of an AP is 10, the sum of their squares is 380. Find the first term and common difference?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!

Let me to decipher the abbreviation  AP.  It is   "Arithmetic Progression".

Let  x  be the  third  term of our arithmetic progression and  d  be its  common difference.
Then the five first terms are

x - 2d
x - d
x
x + d
x + 2d.

The sum of these terms is  5x.  The sum of their squares is 5x%5E2 + 10d%5E2.  It is easy to check.

Thus we have the system of two  (non-linear)  equations

system%285x+=+10%2C%0D%0A5x%5E2+%2B+10d%5E2+=+380%29.

From the first equation we have  x = 10%2F5 = 2.
Substitute it into the second equation.  You will get

5%2A2%5E2 + 10d%5E2 = 380,   or   10d%5E2 = 380+-+20 = 360.

Hence,  d%5E2 = 360%2F10 = 36  and  d = +/-6.

Therefore,  the members of the progression are
-10, -4, 2, 8, 14

- in this order or in the opposite order - it doesn't matter.

You can check that this progression  (or these two progressions)  satisfy to the given conditions.