Understanding Repeating Decimals and Infinite Geometric Series

The video tutorial explains how to convert repeating decimals into simplified fractions by expressing them as infinite geometric series. It covers the formula for the sum of an infinite geometric series and applies it to two examples: 0.222... and 0.151515.... The process involves identifying the first term and common ratio, ensuring the absolute value of the common ratio is less than one, and calculating the sum to find the equivalent fraction.

What is the first step in converting a repeating decimal into a simplified fraction?

• Write it as an infinite geometric series

• Multiply it by 10

• Write it as a finite series

• Convert it directly to a fraction

What is the repeating decimal 0.222... equivalent to in terms of a geometric series?

• 0.2 + 0.2 + 0.2 + ...

• 0.22 + 0.22 + 0.22 + ...

• 0.22 + 0.002 + 0.0002 + ...

• 0.2 + 0.02 + 0.002 + ...

How do you determine the common ratio in a geometric series?

• By subtracting the first term from the second term

• By multiplying the first term by 2

• By dividing the second term by the first term

• By adding the first two terms

What is the common ratio for the series representing 0.222...?

• 1/10

• 1/2

• 1/20

• 1/5

What is the simplified fraction for the repeating decimal 0.222...?

• 2/3

• 1/3

• 1/9

• 2/9

What is the first term in the geometric series for 0.151515...?

• 15/1000

• 3/20

• 1/5

• 15/100

What is the common ratio for the series representing 0.151515...?

• 1/10

• 1/100

• 1/1000

• 1/5

How is the infinite sum calculated for a repeating decimal?

• By subtracting the common ratio from the first term

• By multiplying the first term by the common ratio

• By adding all terms

• By dividing the first term by (1 - common ratio)

What is the simplified fraction for the repeating decimal 0.151515...?

• 15/100

• 1/3

• 5/33

• 3/20

Why is it important that the absolute value of the common ratio is less than 1?

• To make calculations easier

• To ensure the series converges

• To avoid negative numbers

• To simplify the first term