Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.For every integer $ n \geq 1, 1 + 6 + 11 + 16 + \ldots + (5n - 4) = \frac{n(5n - 3)}{2} $.**Proof (by mathematical induction):** Let $ P(n) $ be the equation\[ 1 + 6 + 11 + 16 + \ldots + (5n - 4) = \frac{n(5n - 3)}{2} \]We will show that $ P(n) $ is true for every integer $ n \geq 1 $.**Show that $ P(1) $ is true:** Select $ P(1) $ from the choices below.- $ P(1) = 5 \cdot 1 - 4 $- $ 1 = \frac{1 \cdot (5 \cdot 1 - 3)}{2} $- $ 1 + (5 \cdot 1 - 4) = 1 \cdot (5 \cdot 1 - 3) $- $ P(1) = 1 \cdot \frac{(5 \cdot 1 - 3)}{2} $The selected statement is true because both sides of the equation equal \_\_\_\_\_\_\_\_\_\_.**Show that for each integer $ k \geq 1 $, if $ P(k) $ is true, then $ P(k + 1) $ is true:**Let $ k $ be any integer with $ k \geq 1 $, and suppose that $ P(k) $ is true. The left-hand side of $ P(k) $ is \_\_\_\_\_\_\_\_\_\_, and the right-hand side of $ P(k) $ is \_\_\_\_\_\_\_\_\_\_.*[The inductive hypothesis states that the two sides of $ P(k) $ are equal.]*We must show that $ P(k + 1) $ is true. $ P(k + 1) $ is the equation \( 1

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Description: Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.For every integer $ n \geq 1, 1 + 6 + 11 + 16 + \ldots + (5n - 4) = \frac{n(5n - 3)}{2} $.**Proof (by mathematical induction):** Let \

For, n=1, 1=1(5×1-3)2=1 This statement is true because both sides of the equation are equal.

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rove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2. n(5n - 3) For every integer n2 1, 1 + 6 + 11 + 16 +... + (5n - 4) = 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.

For every integer $ n \geq 1, 1 + 6 + 11 + 16 + \ldots + (5n - 4) = \frac{n(5n - 3)}{2} $.

**Proof (by mathematical induction):** Let $ P(n) $ be the equation

\[ 1 + 6 + 11 + 16 + \ldots + (5n - 4) = \frac{n(5n - 3)}{2} \]

We will show that $ P(n) $ is true for every integer $ n \geq 1 $.

**Show that $ P(1) $ is true:** Select $ P(1) $ from the choices below.

- $ P(1) = 5 \cdot 1 - 4 $
- $ 1 = \frac{1 \cdot (5 \cdot 1 - 3)}{2} $
- $ 1 + (5 \cdot 1 - 4) = 1 \cdot (5 \cdot 1 - 3) $
- $ P(1) = 1 \cdot \frac{(5 \cdot 1 - 3)}{2} $

The selected statement is true because both sides of the equation equal \_\_\_\_\_\_\_\_\_\_.

**Show that for each integer $ k \geq 1 $, if $ P(k) $ is true, then $ P(k + 1) $ is true:**

Let $ k $ be any integer with $ k \geq 1 $, and suppose that $ P(k) $ is true. The left-hand side of $ P(k) $ is \_\_\_\_\_\_\_\_\_\_, and the right-hand side of $ P(k) $ is \_\_\_\_\_\_\_\_\_\_.

*[The inductive hypothesis states that the two sides of $ P(k) $ are equal.]*

We must show that $ P(k + 1) $ is true. $ P(k + 1) $ is the equation \( 1

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