Chapter 12: Q. 27 (page 839)

In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.
$2 + 6 + 18 + . . . . . . . . . . . . . . . . 2 . 3^{n - 1} = 3^{n} - 1$

Short Answer

Expert verified

Given statement is true for all natural numbers.

Step by step solution

Step 1. Given information.

Statement:

$2 + 6 + 18 + . . . . . . . . . . . 2 . 3^{n - 1} = 3^{n} - 1$

Step 2. Show that statement is true for n=1.

$L . H . S . = 2$

$R . H . S . = 3^{1} - 1 = 3 - 1 = 2$

$L . H . S = R . H . S .$

Statement is true.

Step 3. Assume that statement is true for n=k

$2 + 6 + 18 + . . . . . . . . . . . . . . . . . 2 . 3^{k - 1} = 3^{k} - 1$

Step 4. Show that the statement is true for n=k+1.

$L . H . S . = 2 + 6 + 18 + . . . . . . . + 2 . 3^{k - 1} + 2 . 3^{(K + 1) - 1} = 3^{k} - 1 + 2 . 3^{(K + 1) - 1} = 3^{k} - 1 + 2 . 3^{k} = 3^{k} (1 + 2) - 1 = 3^{k} . 3^{1} - 1 = 3^{k + 1} - 1$ , use result of step 3.

$R . H . S . = 3^{K + 1} - 1$

Since $L . H . S . = R . H . S .$ for $n = k + 1$

So statement is true for all natural number.

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In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.
$2 + 6 + 18 + . . . . . . . . . . . . . . . . 2 . 3^{n - 1} = 3^{n} - 1$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Show that statement is true for n=1.

Step 3. Assume that statement is true for n=k

Step 4. Show that the statement is true for n=k+1.

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In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.2+6+18+................2.3n-1=3n-1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Show that statement is true for n=1.

Step 3. Assume that statement is true for n=k

Step 4. Show that the statement is true for n=k+1.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Mechanics Maths

Decision Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.
$2 + 6 + 18 + . . . . . . . . . . . . . . . . 2 . 3^{n - 1} = 3^{n} - 1$