Question-68
Let P(n) be the statement that 13 + 23 + …. + n3 = 
for the positive integer n.
a) What is the statement P(1) ?
P(1) is the statement
b) Show that P (1) is true, completing the basis step of the proof.
The left-hand side of the basis step is ?
The right- hand side of the basis step is ?
c) What is the inductive hypothesis?
The inductive hypothesis is the statement that 13 + 23 + ….+ k3 = ?
d) What do you need to prove in the inductive step?
You need to prove that 1 + 2 +….. + k + 
= ?
e) Complete the inductive step, identifying where you use the inductive hypothesis.
Solution
a) ![1^{3} = [\frac{1.(1+1)}{2}]^{2}](https://mymathangels.com/wp-content/ql-cache/quicklatex.com-3eca332aaf85faa58493d74131822e67_l3.png "Rendered by QuickLaTeX.com")
.
b)
The left-hand side of the basis step is = 1.
The right- hand side of the basis step is = 1.
c)
.
d) 
.
For all k 
1.
e) Ans The inductive hypotheis replaces the quantity 13 + 23 + …. + k3 from the left-hand side of part (d) , which shows that [13 + 23 + … + k3 ] + (k + 1)3.
= 