Question

Let r>1 be real number. Prove by induction that for every integer n ≥ 1,

∑ᵢ=0ⁿ rᶦ=rⁿ+1-1/r-1

Answer 1

To prove the given identity by induction, verify the base case and assume it holds for k, then prove it holds for k + 1.

Step-by-step explanation:

To prove the given identity by induction, we will use the principle of mathematical induction. We will first verify the base case, which is n = 1. Plugging n = 1 into the given equation, we get: ∑i=0¹ ri = r1+1 - 1/r - 1 = r - 1/r - 1 = (r² - 1)/r - 1.

Now, we assume that the given identity holds for some arbitrary positive integer k, and prove that it holds for (k + 1) as well. We have: ∑i=0⁽ᵏ⁺¹⁾ ri = ∑i=0ᵏ ri + rk+1 (using the induction hypothesis) = rk+1+1 - 1/r - 1 + rk+1 = rk+2-1/r-1.

Thus, by the principle of mathematical induction, the given identity is true for all positive integers n ≥ 1.