Understanding and Solving Fraction Series: 1/2 1/6 1/12 1/20 1/30 1/42 1/56 1/72 1/90

Mathematics frequently challenges our understanding through intriguing problem series. One such series involves the addition of fractions, specifically the sequence 1/2, 1/6, 1/12, 1/20, 1/30, 1/42, 1/56, 1/72, 1/90.

Conceptual Breakdown

To solve such a series, we need to understand the underlying pattern and process of addition. The key involves recognizing a pattern within each fraction before we tackle the series in its entirety.

Pattern Identification

Each fraction in the series can be expressed in a simplified form:

1/2 can be written as 2-1/12 1/1 - 1/2 1/6 can be written as 3-2/6 1/2 - 1/3 1/12 can be written as 4-3/12 1/3 - 1/4 1/20 can be written as 5-4/20 1/4 - 1/5 1/30 can be written as 6-5/30 1/5 - 1/6 1/42 can be written as 7-6/42 1/6 - 1/7 1/56 can be written as 8-7/56 1/7 - 1/8 1/72 can be written as 9-8/72 1/8 - 1/9 1/90 can be written as 10-9/90 1/9 - 1/10

By expressing each fraction in this manner, we can see a telescoping effect that allows us to simplify the series significantly.

Step-by-Step Solution

Let's solve the series step by step, starting with the unpacked form:

1/2 (2 - 1)/12 1/1 - 1/2

1/6 (3 - 2)/6 1/2 - 1/3

1/12 (4 - 3)/12 1/3 - 1/4

1/20 (5 - 4)/20 1/4 - 1/5

1/30 (6 - 5)/30 1/5 - 1/6

1/42 (7 - 6)/42 1/6 - 1/7

1/56 (8 - 7)/56 1/7 - 1/8

1/72 (9 - 8)/72 1/8 - 1/9

1/90 (10 - 9)/90 1/9 - 1/10

When we add all of these together, a significant simplification occurs:

1/2 1/6 1/12 1/20 1/30 1/42 1/56 1/72 1/90 (1/1 - 1/2) (1/2 - 1/3) (1/3 - 1/4) (1/4 - 1/5) (1/5 - 1/6) (1/6 - 1/7) (1/7 - 1/8) (1/8 - 1/9) (1/9 - 1/10)

This results in a telescoping series, where most terms cancel out:

(1/1 - 1/2) (1/2 - 1/3) (1/3 - 1/4) (1/4 - 1/5) (1/5 - 1/6) (1/6 - 1/7) (1/7 - 1/8) (1/8 - 1/9) (1/9 - 1/10) 1 - 1/10 9/10

Alternative Calculations Using a Common Denominator

For a more algebraically rigorous approach, we can find a common denominator and sum the fractions. The least common multiple (LCM) can be found by analyzing the prime factorization of the denominators:

2 2^1 6 2^1 x 3^1 12 2^2 x 3^1 20 2^2 x 5^1 30 2^1 x 3^1 x 5^1 42 2^1 x 3^1 x 7^1 56 2^3 x 7^1 72 2^3 x 3^2 90 2^1 x 3^2 x 5^1

The LCM of these numbers is calculated as:

LCM 2^3 x 3^2 x 5^1 x 7^1 2520

Converting each fraction to 2520:

1/2 1260/2520 1/6 420/2520 1/12 210/2520 1/20 126/2520 1/30 84/2520 1/42 60/2520 1/56 45/2520 1/72 35/2520 1/90 28/2520

Adding these numerators gives 2268, so we have:

2268/2520

Simplifying 2268/2520, we use the GCD (36) to get:

63/70 9/10

Thus, the sum of the series is 9/10, confirming the telescoping solution.

Conclusion

This problem demonstrates the elegance of mathematical series and the importance of recognizing patterns. Whether through telescoping or common denominator summation, understanding such series can greatly enhance problem-solving skills in mathematics.