Question

prove by mathematical induction 1cube+2cube+3cube+.......+ncube=nsquare(n+1)square/4​

Answer 1

Step-by-step explanation:

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Answer 2

The statement that 1³+2³ +3³ +.......+n³ =n²(n+1)²/4​ is proved by the Principle of Mathematical Induction.

Given

1³+2³ +3³ +.......+n³ = n²(n+1)²/4​

To Find

Proof of the statement 1³+2³ +3³ +.......+n³ = n²(n+1)²/4​ by Principle of Mathematical Induction

Solution

Statement- 1³+2³ +3³ +.......+n³ = n²(n+1)²/4​

Step 1, n = 1

LHS

1³ = 1

RHS

1² X (1 + 1)²/4

= 1 X 2²/4

4/4

= 1

Therefore, the statement is true for n = 1

Step 2, n = k

Let us assume that the statement holds true for n = k

Hence,

1³ + 2³ + .... + k³ = k²(k+ 1)²/4                         [1]

Step 3, n = k + 1

LHS

= 1³ + 2³ + 3³ + ... + k³ + (k + 1)³

= k²(k+ 1)²/4 + (k + 1)³                                    [from equation [1]]

Taking (k+ 1)² common we get

(k+ 1)²[k²/4 + k + 1]

= (k+ 1)²[(k²+ 4k + 4)/4 ]

= (k+ 1)²[(k²+ 2.2.k + 2²)/4 ]

= (k + 1)²(k + 2)²/4

RHS

(k + 1)²(k + 1 + 1)²/4

= (k + 1)²(k + 2)²/4

Hence LHS = RHS,

The statement that 1³+2³ +3³ +.......+n³ =n²(n+1)²/4​ is proved by the Principle of Mathematical Induction.

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