Chapter 12: Q. 14 (page 839)

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

Short Answer

Expert verified

The statement is proved by mathematical induction.

Step by step solution

Step 1. Given Information

The given expression is $(1 + \frac{1}{1}) (1 + \frac{1}{2}) (1 + \frac{1}{3}) . . . (1 + \frac{1}{n}) = n + 1$

Step 2. Check for n=1.

First, we show that formula is true when $n = 1$ . Because

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

Step 2. Check statement for some k

Next, we assume that formula holds for some k, and we determine whether the formula then holds for $k + 1 .$ We assume that

role="math" localid="1646974240600" $(1 + \frac{1}{1}) (1 + \frac{1}{2}) (1 + \frac{1}{3}) . . . (1 + \frac{1}{k}) = k + 1$

Step 4. show for k+1.

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Check for n=1.

Step 2. Check statement for some k

Step 4. show for k+1.

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Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.1+111+121+13...1+1n=n+1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Check for n=1.

Step 2. Check statement for some k

Step 4. show for k+1.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

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