If the sum of the first 20 terms of the series
$\frac{4.1}{4 + 3. 1 ^{2} + 1 ^{4}} + \frac{4.2}{4 + 3. 2 ^{2} + 2 ^{4}} + \frac{4.3}{4 + 3. 3 ^{2} + 3 ^{4}} + \frac{4.4}{4 + 3. 4 ^{2} + 4 ^{4}} + \dots$
is $\frac{m}{n}$ , where $m$ and $n$ are coprime, then $m + n$ is equal to :-

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TestHub · Accepted Answer

​ r = 1 ∑ 20 ​ 4 + 3 r 2 + r 4 4 r ​ r = 1 ∑ 20 ​ ( r 2 + r + 2 ) ( r 2 − r + 2 ) 4 r ​ 2 r = 1 ∑ 20 ​ ( r 2 − r + 2 1 ​ − r 2 + r + 2 1 ​ ) 2 ( 2 1 ​ − 4 1 ​ ) 4 1 ​ − 8 1 ​ 8 1 ​ − 14 1 ​ ( 382 1 ​ − 422 1 ​ ) = 2 ( 2 1 ​ − 422 1 ​ ) = 422 420 ​ = 211 210 ​ option ( 3 ) ​

Mathematics - Sequence & Series Question with Solution | TestHub

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