Given arithmetic sequence is 9, 15, 21,.
First term = a = 9
Common difference = d =15 9 = 6
(A) For natural number n
nth term = [(n−1)x6]+9=6n+3 where n=1,2,3,
Hence algebraic form of the sequence 9,15,21,. is xn=6n+3.
(B) nth term =6n+3
25th term =6(25)+3=150+3=153
(C) Sum of first n terms is given by,
Sn=12n[X1+Xn]
Sum of 24 terms is
S24=12n[X1+X24]=252[9+147]=1950
Sum of 50 terms is
S50=12n[X1+X50]=502[9+6(50)+3]=7800
Sum of terms from twenty fifth to fiftieth
=S50−S24
=7800−1950
=5850
(D) Let sum of first n terms is 2015,
Sn=12n[X1+Xn]
2015=12n[9+6n+3]
2015=3n2+6n
3n2+6n−2015=0
Solving above equation for n, the value of n is not a natural number.
Hence 2015 can not be the sum of some terms of this sequence.