Question 33525
25th term is a+24d, so a+24d = 75 --eqn1


Sum of 9 = {{{ (n/2)(2a + (n-1)d) }}}
{{{ (9/2)(2a + (9-1)d) = 75 }}}
{{{ 9(2a + 8d) = 150 }}}
{{{ 18a + 72d = 150 }}}

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 {{{ (n/2)(2a + (n-1)d) }}}
{{{ (100/2)(2(-5) + (100-1)(10/3)) }}}
{{{ 50(-10 + (99)(10/3)) }}}
50(-10 + 330)
50(320)
50(320)
16000


jon.