Question

Tìm n\(\in\) N*, biết rằng:

\(\frac{1}{21}+\frac{1}{77}+\frac{1}{165}+.....+\frac{1}{n^2+4n}=\frac{56}{673}\)

nếu giải thích chi tiết mình cho 2 tick

Answer 1

\(\frac{1}{21}+\frac{1}{77}+\frac{1}{165}+...+\frac{1}{n^2+4n}=\frac{56}{673}\)

<=> \(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{n.\left(n+4\right)}=\frac{56}{673}\)

<=> \(4.\left(\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{n.\left(n+4\right)}\right)=4.\frac{56}{673}\)

<=> \(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+...+\frac{4}{n\left(n+4\right)}=\frac{224}{673}\)

<=> \(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{n}-\frac{1}{n+4}=\frac{224}{673}\)

<=> \(\frac{1}{3}-\frac{1}{n+4}=\frac{224}{673}\)

<=> \(\frac{n+4-3}{3.\left(n+4\right)}=\frac{224}{673}\Leftrightarrow\frac{n}{3.\left(n+4\right)}=\frac{224}{673}\)

<=> 673n = 224.3(n+4)

<=> 673n = 224.3.n + 224.3.4

<=> 673n = 672n + 2688

<=> 673n - 672n = 2688

<=> n = 2688