SOLUTION: Simplify the following: <img src="https://i.ibb.co/MPfy9Qc/12.png" height="50px">

Question 1205325: Simplify the following:

Found 3 solutions by ikleyn, math_tutor2020, Edwin McCravy:
Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
.

The original expression is + + + + . . . + = The common term is of the form = , where k is an odd natural number. Therefore, the sequence is reduced to + + + + . . . + = . The sum 1 + 3 + 5 + 7 + . . . + 103 is equal to = . Therefore, the original expression is = 52*26 = 1352. ANSWER. + + + + . . . + = 1352.

Solved.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Tutor @ikleyn probably has the most efficient pathway, but here's another approach.

The pattern in the numerator (1^2,3^2,5^2,7^2,...) follows the sequence (2k-1)^2 where k is an integer and k = 1 is the starting index.
Note how 1,3,5,7,... is an arithmetic sequence.
The highest that k goes is k = 52 because 2k-1 = 2*52 - 1 = 103.

The pattern in the denominator (2,6,10,14,...) is also arithmetic and follows the sequence 4k-2
Note that 4k-2 = 4*52 - 2 = 206.

Each term can be written of the form

We're tasked to find the summation

Use these summation identities

and

and

to have the following steps

Therefore,

Answer by Edwin McCravy(20065)   (Show Source): You can put this solution on YOUR website!

How about this way?: 0.5 + 1.5 + 2.5 + 3.5 + ... + 51.5 = The sum of the whole parts: 0 + 1 + 2 + 3 + ... + 51 = = 1326 PLUS the sum of the 52 decimal parts of 0.5 each: 52 x 0.5 = 26 EQUALS 1352 Edwin

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