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.
$3 + 4 + 5 + . . . . + (n + 2) = \frac{1}{2} n (n + 5)$

Short Answer

Expert verified

We proved this by the principle of mathematical induction

Step by step solution

Given information

We are given a sequence

$3 + 4 + 5 + . . . . + (n + 2) = \frac{1}{2} n (n + 5)$

We check whether the statement is true for n=1

We get,

$\frac{1}{2} n (n + 5) = \frac{1}{2} (6) = 3 = L H S$

The statement is true for n=1

Now consider that the statement is true for n

Hence we have,

$3 + 4 + 5 + . . . . + (n + 2) = \frac{1}{2} n (n + 5)$

Now we prove for n+1

We get,

$3 + 4 + 5 + . . . . + (n + 2) + (n + 3) = \frac{1}{2} (n + 1) (n + 6)$

Consider LHS

$3 + 4 + 5 + . . . . + (n + 2) + (n + 3) = \frac{1}{2} n (n + 5) + (n + 3) = \frac{1}{2} n^{2} + \frac{5 n}{2} + n + 3 = \frac{1}{2} (n + 1) (n + 6)$

Conclusion

We proved this by the principle of 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.
$3 + 4 + 5 + . . . . + (n + 2) = \frac{1}{2} n (n + 5)$

Short Answer

Step by step solution

Given information

We check whether the statement is true for n=1

Now consider that the statement is true for n

Now we prove 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+4+5+....+(n+2)=12n(n+5)

Short Answer

Step by step solution

Given information

We check whether the statement is true for n=1

Now consider that the statement is true for n

Now we prove for n+1

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Decision Maths

Applied Mathematics

Discrete Mathematics

Statistics

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.
$3 + 4 + 5 + . . . . + (n + 2) = \frac{1}{2} n (n + 5)$