Prove the following by using the principle of mathematical induction for all `n in N`:`a+a r+a r^2+dotdotdot+a r^(n-1)=(a(r^n-1))/(r-1)`

Explanation

`P(n):a+ar+ar^(2)+………+ar^(n-1) =(1(r^(n)-1))/(r-1)`
For n=1
`L.H.S. =a R.H.S. = (1(r^(1)-1))/(r-1) =a`
`:. " "L.H.S. =R.H.S.`
`rArr` P(n) is true for n=1
Let P (n) be true for n=k
`:. P(K) : a +ar + ar^(2) +......+ ar^(K-1) =(a(r^(k)-1))/(r-1)`
For n=K+1
`P(k+1) : a + ar + ar^(2) + ....+ ar^(k-1)+ar^(k)`
`=1((r^(k)-1))/(r-1)+ar^(k)`
`=(a[(r^(k)-1)+r^(k)(r-1)]]/(r-1)`
`=(a(r^(k)-1+r^(k+1)-r^(k)))/(r-1)=(1(r^(k+1)-1))/(r-1)`
`rArr` P (n) is also true for n=K+1
Hence from the principle of mathematical induction P (n) is true foa all natural numbers n.