SOLUTION: the sum of the first nine terms of an arithmetic progression is 75 and the twenty-fifth is also 75. find the common difference and the sum of the first hundred terms.

Question 33525: the sum of the first nine terms of an arithmetic progression is 75 and the twenty-fifth is also 75. find the common difference and the sum of the first hundred terms.
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
25th term is a+24d, so a+24d = 75 --eqn1

Sum of 9 =

Scale eqn1 by 3 to give 3a+72d = 225. So we have

3a+72d = 225
18a+72d = 150

Subtract, to give -15a = 75
--> a = -5

So, from a+24d = 75 we get
-5+24d = 75
24d = 80
d = 80/24
d = 10/3

This is:
-5, -5/3, 5/3, 5, 25/3, 35/3, 15, 55/3, 65/3,... which does add up to 75.

So, Sum of 100 terms is

50(-10 + 330)
50(320)
50(320)
16000

jon.

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