Question

Show that the sequence log(a), log(ab), log(ab²), log(ab³), ... is an arithmetic progression (AP). What is the nth term of this sequence?

a) The sequence is not an arithmetic progression.
b) The sequence is an arithmetic progression.
c) The nth term is log(a) + (n - 1)log(b).
d) The nth term is log(a) + nlog(b).

Answer 1

The sequence log(a), log(ab), log(ab²), log(ab³), ... is an arithmetic progression with the nth term being log(a) + (n - 1)log(b).

Step-by-step explanation:

To show that the sequence log(a), log(ab), log(ab²), log(ab³), ... is an arithmetic progression (AP), we need to show that the difference between consecutive terms is constant. Let's calculate the difference between any two consecutive terms:

d = log(ab) - log(a) = log(ab/a) = log(b)

Since the difference d is constant and equal to log(b), we can conclude that the given sequence is an arithmetic progression.

The nth term of an arithmetic progression can be found using the formula:

an = a1 + (n - 1)d

Substituting the values from our sequence into the formula:

an = log(a) + (n - 1)log(b)