Summation by Sigma Method

Q.

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Q.

Find the 10th common term between the arithmetic series 3+7+11+15+... and 1+6+11 +16+....

Q.

Find the sum of the following series upto n terms :

$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$

Q.

The sum of n terms of the sequence 8, 88, 888, .... is:

1. 

2. 

3. 

4. 

Q.

If $a_1,a_2,a_3,....,a_n$ be an AP of non-zero terms. Prove that

$\frac{1}{a_1{a}_2}$+$\frac{1}{a_2{a}_3}$+....+$\frac{1}{a_{n-1}{a}_n}$=

$\frac{n-1}{a_1{a}_n}$

Q. IF $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r},$ then $\frac{t_n}{S_n}$ is equal to

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

Q. Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0,$with $\alpha$ > $\beta$, For all positive integers n, define,
$a_n=\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta },n\ge 1$
$b_1=1$ and $b_n=a_{n-1}+a_{n+1},n\ge 2$.
Then which of the following option is/are correct?

1. $a_1+a_2+a_3+......+a_n=a_{n+2}-1\text{such that}n\ge 1$
2. $\sum _{n=1}^{\infty }\frac{a_n}{{10}^n}=\frac{10}{89}$
3. $\sum _{n=1}^{\infty }\frac{b_n}{{10}^n}=\frac{8}{89}$
4. $b_n={\alpha }^n+{\beta }^n$ such that $n\ge 1$

Q. The sum of first 12 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+........$ is

1. 204.5
2. 126.25
3. 216.5
4. 156.25

Q.

Differentiate $e^x(x^3+\sqrt{x})$.

Q.

Find the sum of all two-digit numbers which when divided by 4 yield 1 as remainder.

Q. If S 1 , S 2 , S 3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that

Q. If $S_n={\sum }_1^{4n}(-1{)}^{\frac{k(k+1)}{2}}{k}^2$. Then, $S_n$ can take value(s)

1. 1056
2. 1088
3. 1120
4. 1332

Q. 2 + 10 + 30 + 68 + 130 + ...

Q. If $S_n={\sum }_1^{4n}(-1{)}^{\frac{k(k+1)}{2}}{k}^2$. Then, $S_n$ can take value(s)

1. 1056
2. 1088
3. 1120
4. 1332

Q.

Find the sum of the first 15 terms of the series 3 + 5 + 7 + 9 +. . . . . . n terms.

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Q. The sum to n terms of the series $\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+....+....$ is
(a) $\sqrt{2n+1}$

(b) $\frac{1}{2}\sqrt{2n+1}$

(c) $\sqrt{2n+1}-1$

(d) $\frac{1}{2}\left|\sqrt{2n+1}-1\right|$

Q. Find the 20^th term and the sum of 20 terms of the series 2 × 4 + 4 × 6 + 6 × 8 + ...

Q.

For a sequence, ${\Sigma }_{r=1}^{100}{a}_r=\alpha ,{\Sigma }_{r=1}^{50}{a}_{2r-1}=\beta .Find{\Sigma }_{r=1}^{50}{a}_{2r}$

Q.

The sum of the first four terms of an AP is 56 and the sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Q.

A series in G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be equal to

1. 2

2. 5

3. 3

4. 4

Q.

In the expansion of ${}^{20}{C}_r$.$4^{\frac{20-r}{3}}{.6}^{-\frac{r}{4}}$

1. the middle term is rational

2. the total no.of irrational terms = 19

3. the total no. of rational terms = 4

4. the total no.of irrational terms = 18

Q. If $\alpha ,\beta and\gamma$ are the real roots of the equation $x^3+5x^2+9x-6=0$. Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$.
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Q. 1.2.4 + 2.3.7 +3.4.10 + ...

Q. $\frac{1}{1.6}+\frac{1}{6.11}+\frac{1}{11.14}+\frac{1}{14.19}+...+\frac{1}{(5n-4)(5n+1)}$

Q. If $\alpha ,\beta and\gamma$ are the real roots of the equation $x^3+5x^2+9x-6=0$. Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$.
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Q. The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$ is

1. 71
2. 96
3. 142
4. 192

Q.

Find the sum of the first n natural numbers.

1. 

2. 

3. 

4. 

Q. The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$ is

1. 71
2. 96
3. 142
4. 192

Q. 8. n (n+1) (n+4).

Q.

If the sum of the n terms of G.P. is S product is P and sum of their inverse is R, than $P^2$ is equal to

1. 

2. 

3. 

4.