Prove the following by using the principle of mathematical induction for all a+ar+ar^2+....+ar^{n-1}

Let us denote the given equality by P(n), i.e.,

\( a+ar+ar^2+....+ar^{n-1}=\dfrac{a(r^n-1)}{r-1}\)

For n=1,

L.H.S.= a

R.H.S. = \( \dfrac{a(r^n-1)}{r-1}=a\)

Therefore, P(n) is true for n=1.

Let us assume that P(k) is true for some positive integer k , i.e.,

\( a+ar+ar^2+....+ar^{k-1}=\dfrac{a(r^k-1)}{k-1}\).........(i)

Now, we have to prove that P(k+1) is also true

Consider

\( \{ a+ar+ar^2+....+ar^{k-1}\}+ar^{(k+1)-1}\)

= \( \dfrac{a(r^k-1)}{r-1}+ar^k\) ...........[Using (i)]

= \( \dfrac{a(r^k-1)+ar^k(r-1)}{r-1}\)

= \( \dfrac{a(r^k-1)+ar^{k+1}-ar^k}{r-1}\)

= \( \dfrac{ar^k-a+ar^{k+1}-ar^k}{r-1}\)

= \( \dfrac{ar^{k+1}-a}{r-1}\)

= \( \dfrac{a(r^{k+1}-1)}{r-1}\)

Therefore, P(k+1) holds whenever P(k) holds.

Hence, the given equality is true for all natural numbers i.e., N by the principle of mathematical induction.