Question

( frac{1}{1 times 3}+frac{1}{3 times 5}+frac{1}{5 times 7}+dots+frac{1}{(2 n-1)(2 n+1)}=frac{n}{2 n+1} )

Answer 1

The given expression is a sum of fractions with a pattern. It simplifies to n/(2n+1).

Step-by-step explanation:

The given expression is a sum of fractions with a pattern. Each fraction has a denominator of (2n-1)(2n+1), where n represents the position of the fraction in the series.

To find a pattern, let's simplify the expression for a few terms:

`(1/3) + (1/15) + (1/35) = 1/3 + (1/3)(1/5) + (1/3)(1/3)(1/7) = (1/3)(1 + 1/5 + (1/5)(1/7))`
From this, we can observe that the expression in the box has a pattern of (1/3)(1 + 1/5 + (1/5)(1/7)), which simplifies to (1/3)(n/(2n+1)).

Hence, the expression in the box is equal to n/(2n+1).