Question -
Answer -
Let P (n): a + ar + ar2 + … + arn– 1 = a [(rn – 1)/(r – 1)]
Let us check for n = 1,
P (1): a = a (r1 – 1)/(r-1)
: a = a
P (n) is true for n = 1.
Now, let us check for P (n) is true for n = k, and have toprove that P (k + 1) is true.
P (k): a + ar + ar2 + … + ark – 1 =a [(rk – 1)/(r – 1)] … (i)
So,
a + ar + ar2 + … + ark – 1 +ark
Now, substituting the value of P (k) we get,
= a [(rk – 1)/(r – 1)] + ark byusing equation (i)
= a[rk – 1 + rk(r-1)] / (r-1)
= a[rk – 1 + rk+1 – r‑k]/ (r-1)
= a[rk+1 – 1] / (r-1)
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.