Question

use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

1 + 5 + 9 + ..... + (4n - 3) = n(2n - 1)

Short answer

We proved that the statement is true using the mathematical induction

Step-by-step solution

Given information

We are given a sequence1 + 5 + 9 + ..... + (4n - 3) = n(2n - 1)

Check whether the statement is true for n=1

We have

$$\begin{gathered} n(2n-1) \\ =1(2-1) \\ =1 \\ =LHS \end{gathered}$$

Hence the statement is true for n=1

Consider that the statement is true for n

Hence we have

$1+5+9+.....+(4n-3)=n(2n-1)$

Now we prove that it is true for n+1 we get,

That is we have to prove that

$1+5+9+...+(4n-3)+(4n+1)=(n+1)(2n+1)$

Consider LHS

$$\begin{gathered} 1+5+9+...+(4n-3)+(4n+1) \\ =n(2n-1)+(4n+1) \\ =2n^2-n+4n+1 \\ =2n^2+3n+1 \\ =(n+1)(2n+1) \end{gathered}$$

Conclusion

We proved that the statement is true using the mathematical induction