The numbers 3^(2sin 2theta-1) , 14 , 3^(4-2sin 2theta) form first three terms of an AP , show that - Maths - Sequences and Series

Question

The numbers 3^(2sin 2theta-1) , 14 , 3^(4-2sin 2theta) form first
three terms of an AP , show that the fifth term is 53

Aarushi Mishra · Accepted Answer

We know that if number a,b and c are in A
P
 then2b=a+cSince 32sin2θ-1, 14 and 34-2sin2θ are in A
P
2×14= 32sin2θ-1+ 34-2sin2θ28=32sin2θ3+ 3432sin2θLet 32sin2θ=t28=t3+ 81t84t=t2+3×81t2-84t+3×81=0t2-81t-3t+3×81=0tt-81-3t-81=0t-3t-81=0t=81 or t=3Note choose t  which forms increasing A
P
  as the fifth term is greater than second, the other will also give the same A
P  but in reverse orderTerms of A P
 are t3, 14,81t, Note choosing t=3 will give increasing A
P
Thus firse three term are 1, 14, 27Common difference if A
P
 d= 14-1=27-14=13 and first term a=1We know that nth temr of A
P   an=a+n-1da5=1+4×13=53

The numbers 3^(2sin 2theta-1) , 14 , 3^(4-2sin 2theta) form first three terms of an AP , show that the fifth term is 53