Mathematical induction

Question

Use mathematical induction to prove that (bottom)r=1 Ʃ (top)n, r In x=(n÷2) In x⁽ⁿ⁺¹⁾ for all n E N

Answer 1

Sum(r) ln x = (n/2) ln (x^(n+1))

Sum(r) ln x = (n/2)(n+1) ln (x)

Sum(r) = n*(n+1)/2 <---- ln(x) cancels for x not equal to 1

Proof:

r=1: 1 = 1*(1+1)/2

r=2: 1 + 2 = 3 = 2*(1+2)/2

r=3: 1+2+3 = 6 = 3*4/2 = 3*(3+1)/2

So the statement holds for n=1,2,3

Suppose the statement holds for some positive

integer k in N, such that k>3

Then the induction hypothesis is:

1+2+3+.....+k = k(k+1)/2

1+2+3+....+k + (k+1) =

k(k+1)/2 + (k+1) = <--- substitutes the induction hypothesis

(k+1) [ k/2 + 1] = <--- factors out k+1

(k+1) [ k+2]/2 <--- common denominator

[end of proof]