Question

Let r ≠ 1 be a real number. Use induction to prove that a ar ar² ⋯ ar^(n−1) = a(1−r^n)/(1−r) for every positive integer n.

a) Base case: n = 1; Inductive step: Assume true for n = k; Prove for n = k+1.
b) Base case: n = 0; Inductive step: Assume true for n = k; Prove for n = k+1.
c) Base case: n = 1; Inductive step: Assume true for n = k; Prove for n = k−1.
d) Base case: n = 0; Inductive step: Assume true for n = k; Prove for n = k−1.

Answer 1

To prove the given statement a ar ar² ⋯ ar^(n−1) = a(1−r^n)/(1−r) using induction, we need to show that the base case holds and then prove the statement for the inductive step.

Step-by-step explanation:

To prove the given statement using induction, we will follow the steps provided.

a) Base case: n = 1;

When n = 1, the equation becomes: a = a(1-r^1)/(1-r). This is true, so the base case holds.

b) Inductive step: Assume true for n = k;

Assume that the statement is true for n = k, which means: a ar ar^2 ... ar^(k-1) = a(1 - r^k)/(1 - r).

c) Prove for n = k+1;

To prove for n = k+1, we need to show that: a ar ar^2 ... ar^(k-1) ar^k = a(1 - r^(k+1))/(1 - r).

d) Prove for n = k-1;

To prove for n = k-1, we need to show that: a ar ar^2 ... ar^(k-1) = a(1 - r^(k-1))/(1 - r).