Chapter 12: Q. 4 (page 830)

use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
3 + 5 + 7 +........ + (2n + 1) = n(n + 2)

Short Answer

Expert verified

We proved that the statement is true for all natural numbers using Mathematical induction

Step by step solution

Given information

We are given a sequence3 + 5 + 7 +........ + (2n + 1) = n(n + 2)

Check for n=1

We have,

$n (n + 2) = 1 (1 + 2) = 3 = L H S$

The statement is true for n=1

Consider the statement is true for n

Hence we get,

$3 + 5 + 7 + . . . . . . . . + (2 n + 1) = n (n + 2)$

Now prove that the statement is true for n+1

That is prove that

$3 + 5 + 7 + . . . . + (2 n + 1) + (2 n + 3) = (n + 1) (n + 3)$

Consider LHS

$3 + 5 + 7 + . . . . . . . . + (2 n + 1) + (2 n + 3) = n (n + 2) + (2 n + 3) = n^{2} + 2 n + 2 n + 3 = (n + 1) (n + 3)$

Conclusion

We proved that the statement is true for all natural numbers using Mathematical induction

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use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
3 + 5 + 7 +........ + (2n + 1) = n(n + 2)

Short Answer

Step by step solution

Given information

Check for n=1

Consider the statement is true for n

Now prove that the statement is true for n+1

Conclusion

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use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 3 + 5 + 7 +........ + (2n + 1) = n(n + 2)

Short Answer

Step by step solution

Given information

Check for n=1

Consider the statement is true for n

Now prove that the statement is true for n+1

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Statistics

Probability and Statistics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
3 + 5 + 7 +........ + (2n + 1) = n(n + 2)