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Chapter 12: Q. 2 (page 830)

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

Expert verified

We proved that the statement is true using the mathematical induction

Step by step solution

01

Given information

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

02

Check whether the statement is true for n=1

We have

n(2n-1)=1(2-1)=1=LHS

Hence the statement is true for n=1

03

Consider that the statement is true for n

Hence we have

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

04

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

1+5+9+...+(4n-3)+(4n+1)=n(2n-1)+(4n+1)=2n2-n+4n+1=2n2+3n+1=(n+1)(2n+1)

05

Conclusion

We proved that the statement is true using the mathematical induction

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In Problems 61鈥�70, express each sum using summation notation.

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