Question

The sum of the first 7terms of ap is 63 and sum of its next 7terms is 161find the 28th term of this ap

Answer 1

Sum of first seven terms,
T(1)+T(2)+T(3)+......+T(7)=S(7)=63 - - - - - - (1)
Sum of next seven terms,
T(8)+T(9)+T(10)+......+T(14)=S(14)-S(7)=161 - - - - (2)
Substituting value of S(7) in equation (2),
S(14)-63=161
S(14)=161+63=224 - - - - - - - - (3)


S(n) = n/2 × [2a+(n-1)d]
Equation (1) implies,
S(7)=7/2 ×[2a+6d] = 63
7 × [a+3d]=63
a+3d=63/7
a+3d=9
a=9-3d - - - - - - - - - - - (4)

Equation (3) implies,
S(14)=14/2 ×[2a+13d] = 224
7 × [2a+13d]=224
2a+13d=224/7
2a+13d=32 - - - - - - - - - - (5)
Substituting (4) in (5) we get,
2(9-3d)+13d=32
18-6d+13d=32
18+7d=32
7d=32-18
7d=14
d=14/7
d=2
Substituting value of d in (4)
a=9-3d
a=9-3(2)
a=9-6
a=3

Hence a=3 & d=2 for given AP
T(n)=a+(n-1)d
T(28)=3+(28-1)*2
=3+(27*2)
=3+54
=57

Hence 28th term of given AP is 57