displaystyle ∑ r = 1n (r/r4 + r2 + 1) is equal to

Question

displaystyle ∑ r = 1n (r/r4 + r2 + 1) is equal to (A) (n2 + n/2 (n2 + n + 1)) (B) (n2 + 2 n/2 (n2 + n + 1)) (C) (2 n2 + n/2 (n2 + n + 1)) (D) (n2 + n/(n2 + n + 1))

Tardigrade Admin · Accepted Answer

Let, Î(r)=(r/r4 + r2 + 1)=(r/(r2 + r + 1) (r2 - r + 1)) =(1/2) ((r2 + r + 1) - (r2 - r + 1)/(r2 + r + 1) (r2 - r + 1)) =-(1/2)((1/r2 + r + 1) - (1/r2 - r + 1)) .=-(1/2)(V r)-V(r-1)) ⇒ displaystyle∑r=1n I (r)=-(1/2)(V(n)-V(0)) =-(1/2)((1/n2+n+1)-1) =(1/2)(1-(1/n2+n+1))=(1/2) ((n2+n)/(n2+n+1))

Q. $r = 1 \sum n \frac{r}{r ^{4} + r ^{2} + 1}$ is equal to

$\frac{n ^{2} + n}{2 ( n ^{2} + n + 1 )}$

$\frac{n ^{2} + 2 n}{2 ( n ^{2} + n + 1 )}$

$\frac{2 n ^{2} + n}{2 ( n ^{2} + n + 1 )}$

$\frac{n ^{2} + n}{( n ^{2} + n + 1 )}$

Q. r=1∑n​r4+r2+1r​ is equal to

2(n2+n+1)n2+n​

2(n2+n+1)n2+2n​

2(n2+n+1)2n2+n​

(n2+n+1)n2+n​

Let, I(r)=r4+r2+1r​=(r2+r+1)(r2−r+1)r​ =21​{(r2+r+1)(r2−r+1)(r2+r+1)−(r2−r+1)​} =−21​(r2+r+11​−r2−r+11​) =−21​(Vr)−V(r−1)) ⇒r=1∑n​I(r)=−21​(V(n)−V(0)) =−21​(n2+n+11​−1) =21​(1−n2+n+11​)=21​(n2+n+1)(n2+n)​

Q. $r = 1 \sum n \frac{r}{r ^{4} + r ^{2} + 1}$ is equal to

$\frac{n ^{2} + n}{2 ( n ^{2} + n + 1 )}$

$\frac{n ^{2} + 2 n}{2 ( n ^{2} + n + 1 )}$

$\frac{2 n ^{2} + n}{2 ( n ^{2} + n + 1 )}$

$\frac{n ^{2} + n}{( n ^{2} + n + 1 )}$