Prove by mathematical induction: 1³ + 2³ + ... + n³ = [n(n+1)/2]²

Question

Prove by mathematical induction that for all positive integers $n$ :

1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2

(NCERT Class 11, Exercise 4.1)

Solution — Step by Step

LHS: $1^3 = 1$

RHS: $\left[\frac{1 \cdot 2}{2}\right]^2 = 1^2 = 1$

LHS = RHS ✓. The statement holds for $n = 1$ .

Assume that for some positive integer $k$ :

1^3 + 2^3 + \cdots + k^3 = \left[\frac{k(k+1)}{2}\right]^2

We need to prove it holds for $n = k + 1$ .

We need to show: $1^3 + 2^3 + \cdots + k^3 + (k+1)^3 = \left[\frac{(k+1)(k+2)}{2}\right]^2$

Start with the LHS. Using the inductive hypothesis:

\text{LHS} = \left[\frac{k(k+1)}{2}\right]^2 + (k+1)^3

Factor out $(k+1)^2$ :

= (k+1)^2\left[\frac{k^2}{4} + (k+1)\right]

= (k+1)^2\left[\frac{k^2 + 4k + 4}{4}\right]

= (k+1)^2 \cdot \frac{(k+2)^2}{4}

= \left[\frac{(k+1)(k+2)}{2}\right]^2 = \text{RHS}

By the principle of mathematical induction, the statement is true for all positive integers $n$ .

\boxed{1^3 + 2^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2}

Why This Works

Mathematical induction works like a chain of dominoes. The base case knocks down the first domino. The inductive step proves that if any domino falls, the next one must fall too. Together, they guarantee all dominoes fall.

The beautiful identity itself says: the sum of cubes equals the square of the sum. That is, $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ . This is sometimes called Nicomachus’s theorem, known for over 2000 years.

Alternative Method — Direct algebraic proof

Without induction, use the telescoping identity:

(k+1)^4 - k^4 = 4k^3 + 6k^2 + 4k + 1

Sum from $k = 1$ to $n$ : the LHS telescopes to $(n+1)^4 - 1$ . The RHS becomes $4\sum k^3 + 6\sum k^2 + 4\sum k + n$ . Substitute known formulas for $\sum k$ and $\sum k^2$ , then solve for $\sum k^3$ . This gives the same result but without induction.

For CBSE boards, induction proofs follow a rigid structure: base case, hypothesis, inductive step, conclusion. Write each as a clearly labelled paragraph. The most marks are in Step 3 — the algebraic manipulation. Make sure your factoring is clean and each line follows logically from the previous one.

Common Mistake

The trickiest algebra is in Step 3: factoring $(k+1)^2$ from $\frac{k^2(k+1)^2}{4} + (k+1)^3$ . Students often expand everything and get lost in a mess of terms. The cleaner approach: factor out $(k+1)^2$ before expanding. This keeps the expression manageable and leads directly to $\frac{(k+2)^2}{4}$ inside the brackets.

Question

Solution — Step by Step

Why This Works

Alternative Method — Direct algebraic proof

Common Mistake

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