For the series 3, 8 15, 24 , 35… what is the difference of the sum of the first 25 terms and the first 75 terms.

The terms of the given series is 3,8,15,24,
35....

a1 = 3, a2 = 8, a3 = 15, a4 = 24, a5
=35

Therefore we notice that an =  (n+1)^2-1 = n^2+2n =
n(n+1)+n.

Therefore  the sum  Sn of the n terms of the
series  could be written  as:

Sn = (1*2+1)+(2*2+2)
+(3*4+3)+(4*5+4)+(5*6+4)+.....

Sn = sum (n)(n+1) + sum of
(1+2+3+..)

Sn = sum n(n+1) +
n(n+1)/2

Sn = (n+2)(n+1)(n)/3
+n(n+1)/2

Sn = n(n+1){2(n+2)+3}/6
.

Sn = n(n+1)(2n+7)/6.

Tally:
 S1 = 1*(1+1)(2*1+7) / 6 = 18/6 = 3. Also S1 =a1 = 3.

S2 =
2(2+1)(2*2+7)/6 = 11.  Also S2 = a1+ a2 = 3+ 8 = 11.

S3 =
3(3+1)(2*3+7)/6 = 26, Also S3 = a1+a2+a3 = 3+8+15 =
26.

Therefore the sum of 25 terms S25 = 25(26)(2*25+7)/6 =
6175.

Similarly S75 = (75)(76)(2*75)/6 =
149150.

Therefore S75 - S25 = 149150-6175 =
142975.

Therefore the difference of the sum of the first 25
terms and the sum of the first 75 tems of the series is S75 - S25 =
142975.