Prove by mathematical induction
\[
3^n \gt n^2 \quad \text{for} \quad n \ge 2
\]
Prove the statement is true for a particular n. Here n = 2 is an obvious choice.
To show this is true for n=2, observe that the left hand side = 9 and the right hand side = 4.
So the statement is true for n=2.
Assume the statement is true for n = k and show it is true for n = k + 1.
To show the statement is true for n = k+1, show \( LHS - RHS \gt 0 \). So
\[ \begin{align}
LHS - RHS &= 3^{k+1} - (k+1)^2 \\
&= 3.3^k - (k^2 + 2k + 1) \quad (1)
\end{align} \ \]
Use the inductive hypothesis to simplify the right hand side of equation 1.
By the inductive hypothesis,
\( \quad 3.3^k - (k^2 + 2k + 1) \)
\( \quad \gt 3k^2 - (k^2 + 2k + 1) \)
\[ \begin{align}
&= 2k^2 - 2k - 1 \\
&= 2(k^2 - k - \frac{1}{2}) \\
&= 2[(k - \frac{1}{2})^2 - \frac{3}{4}] \quad \text{(by completing the square)}\\
&\gt 0 \quad \text{(for } k \ge 2)
\end{align} \ \]
So the statement is true for n = k + 1.
By the principle of mathematical induction,
The statement is true for \( n \ge 2 \).
Important This is not a 'textbook on a screen' - this is an interactive learning system. Which means you have to 'act' to make it go.
Look through the options below and select the one you think is the best response. You will gain most benefit if you work the problem with pen and paper as you go, i.e. decide what you think is the correct option and write down what you think the result will be before you click. If the next step doesn't display, click another option.