Show that the sequence `loga ,log(a b),log(a b^2),log(a b^3),` is an A.P. Find its nth term.

Explanation

We have,
log(ab)-loga=log`((ab)/a)=logb`
`log(ab^(2))-log(ab)=log((ab^(2))/(ab))=logb`
`log(ab^(3))-log(ab^(2))=log((ab^(3))/(ab^(2)))=log b`
It follows from the above results that the difference of a term and the preceding term is always same. So, the given sequence is an A.P with common difference log b. Now,
`a_(n)=a+(n-1)d`
`=loga+(n-1)logb`
`=loga+logb^(n-1)`
`=log(ab^(n-1))`