Sigma n

Q. The sum to $n$ terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots \dots$ is $\frac{1}{2}-\frac{1}{a(n^b+n^c+d{n}^4+1)}$, then $a+b+c+d=$

1. $5$
2. $4$
3. $8$
4. $6$

Q. The value of positive integer $n$ for which the quadratic equation, $\sum _{k=1}^n(x+k-1)(x+k)=10n$ has solutions $\alpha$ and $\alpha +1$ for some $\alpha \in R$, is

Q. If $(x^2+x+1)+(x^2+2x+3)+(x^2+3x+5)+\cdots +(x^2+20x+39)=4500,$ then $x$ is equal to

1. $10$
2. $-10$
3. $20.5$
4. $-20.5$

Q. If $S_1,S_2,S_3\dots \dots ,S_n,\dots$ are the sums of infinite geometric series whose first terms are $1,2,3,\dots \dots n,\dots \dots$ and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \dots \frac{1}{n+1}\dots \dots$ respectively, then the value of $\sum _{r=1}^{2n-1}{S}_r^2$ is

1. $\frac{n(n+1)(2n+1)}{2}-1$
2. $\frac{n(n+1)(n+2)}{2}-1$
3. $\frac{n(2n+1)(4n+1)}{3}-1$
4. $\frac{n(2n+1)(4n+1)}{3(n+1)}-1$

Q. The $n^{th}$ term of series $\frac{1}{1}+\frac{1+2}{2}+\frac{1+2+3}{3}+....$ will be
[AMU 1982]

1. $\frac{n+1}{2}$
2. $\frac{n-1}{2}$
3. $\frac{n^2+1}{2}$
4. $\frac{n^2-1}{2}$

Q. If $\underset{i-1}{\sum ^n}i=\frac{n(n+1)}{2},then\underset{i-1}{\sum ^n}(3i-2)=$

1. $\frac{n(3n-1)}{2}$
2. $\frac{n(3n+1)}{2}$
3. n(3n + 2)
4. $\frac{n(3n+1)}{4}$