Evaluate the following sequence of fractions : What do you observe? Can you explain why this happens? Class 8

Evaluate the following sequence of fractions : What do you observe? Can you explain why this happens? Class 8

Question 1.

Evaluate the following sequence of fractions : Class 8

[latex]\frac{1}{3}[/latex], [latex]\frac{(1+3)}{(5+7)}[/latex], [latex]\frac{(1+3+5)}{(7+9+11)}[/latex]

What do you observe? Can you explain why this happens?

[Hint : Recall what you know about the sum of the first n odd numbers.]

Solution:

It is [latex]\frac{1}{3}[/latex], [latex]\frac{1}{3}[/latex], [latex]\frac{1}{3}[/latex], ...

Here [latex]\frac{1}{3}[/latex] = [latex]\frac{1^2}{2^2-1}[/latex] = [latex]\frac{1}{3}[/latex]

[latex]\frac{1+3}{5+7}[/latex] = [latex]\frac{2^2}{4^2-2^2}[/latex] = [latex]\frac{2 \times 2}{(4+2)(4-2)}[/latex] = [latex]\frac{1}{3}[/latex]

[latex]\frac{1+3+5}{7+9+11}[/latex] = [latex]\frac{3^2}{6^2-3^2}[/latex] = [latex]\frac{3 \times 3}{(6+3)(6-3)}[/latex]

= [latex]\frac{1}{3}[/latex] and so on.