Question

If a,b,c are in AP then show that 3^a , 3^b , 3^c are in GP

Answer 1

If a , b and c are in AP

so they can be written as b-d , b , b+d where d is common difference of the AP

For a sequence to be in GP the ratio of subsequent terms should be equal

i.e. ratio of 1st term to 2nd term = ratio of 2nd term to 3rd term

in this case

ratio of 1st term to 2nd term

3^a/3^b = 3^(a-b) = 3^( b - d - b ) = 3^(-d)

ratio of 2nd term to 3rd term

3^b/3^c = 3^(b-c) = 3^( b - b - d ) = 3^(-d)

hence proved