In an AP, if Sn= 3n2+5n and ak=164, find the value of k.

Problem : In an AP, if S_n= 3n²+5n and a_k=164, find the value of k.

Doubt by Muskan

Solution : 

S_n= 3n²+5n
Putting n=1
S₁=a₁=3(1)²+5(1)
S₁=a₁=3+5
S₁=a₁=8

k=?
S_n= 3n²+5n
Putting n=k
S_k=3k²+5k

a_k=164 (Given)

S_n=n/2[a+a_n]
S_k=k/2[a+a_k]

3k²+5k=k/2[8+164]
3k²+5k=k/2[172]
3k²+5k=86k
3k²+5k-86k=0
3k²-81k=0
k(3k-81)=0
k=0 or 3k-81=0
k=0 or 3k=81
k=0 or k=81/3
k=0 or k=27

But k≠0
Hence, k=27