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Chapter 12: Q. 3 (page 830)

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

3+4+5+....+(n+2)=12n(n+5)

Short Answer

Expert verified

We proved this by the principle of mathematical induction

Step by step solution

01

Given information

We are given a sequence

3+4+5+....+(n+2)=12n(n+5)

02

We check whether the statement is true for n=1

We get,

12n(n+5)=12(6)=3=LHS

The statement is true for n=1

03

Now consider that the statement is true for n

Hence we have,

3+4+5+....+(n+2)=12n(n+5)

04

Now we prove for n+1

We get,

3+4+5+....+(n+2)+(n+3)=12(n+1)(n+6)

Consider LHS

3+4+5+....+(n+2)+(n+3)=12n(n+5)+(n+3)=12n2+5n2+n+3=12(n+1)(n+6)

05

Conclusion

We proved this by the principle of mathematical induction.

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