1/1.3 1/3.5 1/5.7 ... 1/(2n-1)(2n 1) = n/2n 1?
let me fix it for you
should say:
1/(1*3) + 1/(3*5) + 1/(5*7) + ... + 1/(2n-1)(2n + 1) = n/(2n + 1)
Now what about it ???
Are you trying to prove it is true?
Try Induction.
1/(1*3) + 1/(3*5) + 1/(5*7) + ... + 1/(2n-1)(2n + 1) = n/(2n + 1)
To determine whether the expression 1/(2n-1)(2n+1) is equal to n/(2n+1), we can simplify both expressions and compare them.
Let's simplify the expression 1/(2n-1)(2n+1):
By applying the concept of multiplication in algebra, we can rewrite it as 1/[(2n-1)*(2n+1)].
To simplify further, we can use the distributive property to multiply (2n-1) with (2n+1):
1/[(2n)*(2n) + 2n - 2n - 1*(2n+1)].
This simplifies to:
1/(4n^2 + 2n - 2n -1),
which further reduces to:
1/(4n^2 - 1).
Now let's simplify the expression n/(2n+1):
This expression is already in its simplest form, so we do not need to perform any additional steps.
Now we can compare the simplified expressions:
1/(4n^2 - 1) vs. n/(2n+1).
By observing both expressions, we can see that they are not equal. Therefore, the expression 1/(2n-1)(2n+1) is not equivalent to n/(2n+1).
I have explained how to get the answer to this question by simplifying both expressions and comparing them.