Chapter 12: Q 16. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$- 2 - 3 - 4 - . . . - (n + 1) = - \frac{1}{2} n (n + 3)$

Short Answer

Expert verified

The statement is shown for all the natural numbers.

Step by step solution

Step 1. Given information

The statement is $- 2 - 3 - 4 - . . . - (n + 1) = - \frac{1}{2} n (n + 3)$

Step 2. Show that the given statement is true for all natural numbers.

Put $n = 1$ in the given statement.

$- 2 = = - \frac{1}{2} (1) (1 + 3) - 2 = - 2$

Next, we assume that the above statement is true for some $k$ . so that localid="1647674105542" $- 2 - 3 - 4 \dots - (k + 1) = - \frac{1}{2} k (k + 3)$

For width="61">n=k+1

$\begin{array}{rcl} - \frac{1}{2} k (k + 3) - (k + 2) & = & = - (\frac{k^{2} + 3 k + 2 (k + 2)}{2}) \\ = & - (\frac{k^{2} + 3 k + 2 k + 4}{2}) \end{array}$

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$- 2 - 3 - 4 - . . . - (n + 1) = - \frac{1}{2} n (n + 3)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is true for all natural numbers.

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.-2-3-4-...-(n+1)=-12n(n+3)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is true for all natural numbers.

One App. One Place for Learning.

Most popular questions from this chapter

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Statistics

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Study anywhere. Anytime. Across all devices.

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$- 2 - 3 - 4 - . . . - (n + 1) = - \frac{1}{2} n (n + 3)$