Question

n> 1. n(n + 1) 72 for all integers n > 1. 11. 1° +2° + ... +n3 =

prove by mathematical induction

Answer 1

P(n) => 1³+2³+3³+......+n³ =

n'in +1)

step 1

let n=1

LSH -> 1³ =1,

RHS->
1²(1+1)
  
- -

LSH=RHS

P(1) is true

Step2

Let P(n) be true for n=k then

13 +2° +33 + ... + = ...equation1

We need to show that k+1 is also true

1+2+3+..+ (R+1)*(x+1+1) + (+1) =.

1+2+3+..+ + (+1)= = : (k+1)(c+2)*

1+2+3+..+ (R+1)*(: +4 +4) + (+1) =.

k(k+ 1)2 + 4(k+1) 13 +2° +33 + ... + k + (k+1) =

化+1} 1+2+3+..+ +(k+1)= = + (+1)

As we see both sides (k+1)³ is added

and LHS=RHS by equation 1

So P(k+1) is true by induction method