7.1 Q-9

Exercise 7.1 — Question 9

Prove by Mathematical Induction that for all positive integers $n$:

$1^3+2^3+3^3+\cdots +n^3=\frac{n^2(n+1{)}^2}{4}$

---

Step 1: Base Case ($n=1$)

Left-hand side (LHS): $1^3=1$

Right-hand side (RHS): $\frac{1^2(1+1{)}^2}{4}=\frac{1\cdot 4}{4}=1$

Since LHS $=$ RHS $=1$, the statement is true for $n=1$.

---

Step 2: Inductive Hypothesis

Assume the statement is true for $n=k$, i.e., assume:

$1^3+2^3+3^3+\cdots +k^3=\frac{k^2(k+1{)}^2}{4}$

---

Step 3: Inductive Step

We must prove the statement is true for $n=k+1$, i.e., we need to show:

$1^3+2^3+\cdots +k^3+(k+1{)}^3=\frac{(k+1{)}^2(k+2{)}^2}{4}$

Starting from the LHS:

$\underset{\text{use inductive hypothesis}}{\underbrace{1^3+2^3+\cdots +k^3}}+(k+1{)}^3$

$=\frac{k^2(k+1{)}^2}{4}+(k+1{)}^3$

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

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

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

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

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

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

This is exactly the RHS for $n=k+1$. ✓

---

Conclusion

By the Principle of Mathematical Induction, the statement $1^3+2^3+3^3+\cdots +n^3=\frac{n^2(n+1{)}^2}{4}$ is true for all positive integers $n$.