For a given geometric sequence the 7 th term a7 is equal to

For a given geometric sequence, the 7 th term, a7 , is equal to 414 , and the 10 th term, a10 , is equal to 82 . Find the value of the 14 th term, a14 . If applicable, write your answer as a fraction.

Solution

Let the first term of geometric series is \'a\' and common ratio is \'r\'

Then we know nth term tn=ar^(n-1)

Given a7=414=ar^6..............(1)

a10=82=ar^9..................(2)

Equation (1)/(2)

(ar^9)/(ar^6)=82/414

=> r^3=41/207

=> r=(41/207)^(1/3)

therefore a7=414=ar^6=a(41/207)^(6/3)=a(41/207)^2

=> a=414(207/41)^2

therefore 14th term= a14=ar^13=414(207/41)^2*(41/207)^(13/3)=414(41/207)^(7/3)