Show that for each integer k ≥ 1, if P(k) is true, then P(k + 1) is true. Look at the equation in Description below, what's the right side equal to?

Question

We must show that P(k + 1) is true. What is the right side equal to? Give me a correct answer.

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

Answer 1

Note that for any integer M, 1 + 1/M = (M+1)/M

n=1 : 1 + 1/1 = 2

n=2: 2/1 * 3/2 = 3

n=3: 2/1 * 3/2 * 4/3 = 4

n=4: 2/1 * 3/2 * 4/3 * 5/4 = 5

n=5: 2/1 * 3/2 * 4/3 * 5/4 * 6/5 = 6

.....

n=k: 2/1 * 3/2 * 4/3 * 5/4 * 6/5 * .... * k/(k-1) * (k+1)/k = k+1 for integer k>5

and this statement serves as the induction hypothesis

So to prove the statement holds for N=k+1,

2/1 * 3/2 * 4/3 * 5/4 * 6/5 * ..... * k/(k-1) * k+1/k * ( k+2)/(k+1) =

[2/1 * 3/2 * 4/3 * 5/4 * 6/5 * ..... * k/(k-1) * k+1/k ]* ( k+2)/(k+1)

[k+1] * (k+2)/(k+1) = <--- substitutes induction hypothesis

k+2 <-- k+1 factor cancels out

end of proof