The sum displaystyle∑n=121 (3/(4 n-1)(4 n+3)) is equal to

Question

The sum displaystyle∑n=121 (3/(4 n-1)(4 n+3)) is equal to (A) (7/87) (B) (7/29) (C) (14/87) (D) (21/29)

Tardigrade Admin · Accepted Answer

displaystyle∑n=121 (3/(4 n-1)(4 n+3)) =(3/4) displaystyle∑n=121 ((4 n+3)-(4 n-1)/(4 n-1)(4 n+3)) =(3/4) displaystyle∑n=121 (1/4 n-1)-(1/4 n+3) =(3/4)((1/3)-(1/7)+(1/7)-(1/11)+(1/11)- ldots .+(1/83)-(1/87)) =(3/4)((1/3)-(1/87))=(7/29)

Q. The sum $n = 1 \sum 21 \frac{3}{( 4 n - 1 ) ( 4 n + 3 )}$ is equal to

$\frac{7}{87}$

$\frac{7}{29}$

$\frac{14}{87}$

$\frac{21}{29}$

Q. The sum n=1∑21​(4n−1)(4n+3)3​ is equal to

877​

297​

8714​

2921​

n=1∑21​(4n−1)(4n+3)3​ =43​n=1∑21​(4n−1)(4n+3)(4n+3)−(4n−1)​ =43​n=1∑21​4n−11​−4n+31​ =43​(31​−71​+71​−111​+111​−….+831​−871​) =43​(31​−871​)=297​

Q. The sum $n = 1 \sum 21 \frac{3}{( 4 n - 1 ) ( 4 n + 3 )}$ is equal to

$\frac{7}{87}$

$\frac{7}{29}$

$\frac{14}{87}$

$\frac{21}{29}$