how does 4[k+1] raise to 3 chANGES TO 4[K+1] in the next step where does raise to 3 go how - Maths - Principle of Mathematical Induction

Question

how does 4[k+1] raise to 3 chANGES TO 4[K+1] in the next step
where does raise to 3 go how is it solved?in question exercise4 1
Q2

Manbar Singh · Accepted Answer

Here is the complete solution

Let Pn: 13 + 23 + 33 +
+n3 = nn+122 be the given statement
Put n = 1, we getLHS = 13 = 1  ;   RHS = 11+122 = 1Hence, LHS = RHSThus, P1 is true
Let Pk is true
Now,Pk:13 + 23 + 33 +
+k3 = kk+122  Now,13 + 23 + 33 +
+k3 + k+13= kk+122  +  k+13=k2 k+124 + k+13=k2k+12 + 4k+134=k+12k2 + 4k+14    On taking k+12 common in the numerator=k+12k2 + 4k + 44=k+12k2 + 2k2 + 224=k+12k+224=k+1k+1 + 122⇒Pk+1 is true whenever Pk is true
Thus, P1 and Pk+1 is true whenever Pk is true
Thus, by PMI, Pn is true for all n∈N

how does 4[k+1] raise to 3 chANGES TO 4[K+1] in the next step where does raise to 3 go how is it solved?in question exercise4.1 Q2