Sequences and Series Test

Result

Sequences and Series Test - 57

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Weekly Quiz Competition

Question 1
1 / -0

The sum to infinity of the series:
$$\dfrac {3}{{1}^{3}}+\dfrac {5}{{1}^{3}+{2}^{3}}+\dfrac {7}{{1}^{3}+{2}^{3}+{3}^{3}}+..$$ is-

A
5
B

4
C
6

D
1

Solution
Question 2
1 / -0

Sum to $$n$$ terms of the series$$\dfrac { 1 }{ 1.2.3.4 } +\dfrac { 1 }{ 2.3.4.5 } +\dfrac { 1 }{ 3.4.5.6 } +..........$$, is

A
$$\dfrac { { n }^{ 3 } }{ 3\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }$$

B
$$\dfrac { { n }^{ 3 }+{ 6n }^{ 2 }-3n }{ 6\left( n+2 \right) \left( n+3 \right) \left( n+4 \right) }$$

C
$$\dfrac { 15{ n }^{ 2 }+7n }{ 4n\left( n+1 \right) \left( n+5 \right) }$$

D
$$\dfrac { { n }^{ 3 }+6{ n }^{ 2 }+11n }{ 18\left( n+1 \right) \left( n+2 \right) \left( n+3 \right) }$$

Solution
Question 3
1 / -0

If $$\dfrac{x}{0.2} + \dfrac{x}{0.3} + \dfrac{x}{0.6} + \dfrac{x}{0.4} + \dfrac{x}{0.5} = 87$$, then the value of x is equal to

A
0

B
4

C
6

D
1

Solution
Question 4
1 / -0

The value of $$\displaystyle\sum^{100}_{r=2}\dfrac{3^r(2-2r)}{(r+1)(r+2)}$$ is equal to?

A
$$\dfrac{1}{2}-\dfrac{3^{100}}{100(101)}$$

B
$$\dfrac{3}{2}-\dfrac{3^{101}}{101(102)}$$

C
$$\dfrac{3}{2}-\dfrac{3^{100}}{100(101)}$$

D
None of these

Solution

$$ \displaystyle S= \sum_{r = 2}^{100} \dfrac{3^{r}(2-2r)}{r(r+1)(r+2)}$$
$$ \displaystyle S=\sum_{r = 2}^{100} \dfrac{3^{r}}{r(r+2)}-\dfrac{3^{r}+1}{(r+1)(r+2)}$$
$$ = \dfrac{3^{2}}{2(3)} - \dfrac{3^{3}}{3 \times 4}+ \dfrac{3^{3}}{3 \times 4} - \dfrac{3^{4}}{4\times 5}+\dfrac{3^{4}}{4\times 5}+...+\left ( \dfrac{3^{100}}{100 \times 101}-\dfrac{3^{101}}{101 \times 102} \right ) $$
$$ = \dfrac{3^{2}}{2(3)} - \dfrac{3^{101}}{101 \times 102 }$$
$$ S = \dfrac{3}{2} - \dfrac{3^{101}}{101 \times 102}$$
Question 5
1 / -0

If $$f(x)=\displaystyle\Pi^3_{i=1}(x-a_i)+\displaystyle\sum^3_{i=1}a_i-3x$$ where $$a_i < a_{i+1}$$ for $$i=1, 2,$$ then $$f(x)=0$$ has:

A
Only one distinct real root

B
Exactly two distinct real root

C
Exactly $$3$$ distinct real roots

D
$$3$$ equal real roots

Solution
Question 6
1 / -0

A
$$6$$

B
$$7$$

C
$$8$$

D
$$9$$

E
$$10$$

Solution
Question 7
1 / -0

Find sum of series:
$$1.3.5+3.5.7+5.7.9.....$$ ?

A
$$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{8}+\frac{15}{8}$$

B
$$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{18}+\frac{15}{18}$$

C
$$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{20}+\frac{15}{8}$$

D
$$S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{6}+\frac{1}{8}$$

Solution
Question 8
1 / -0

The sum of the series $$1+\dfrac{1}{1!}.\dfrac{1}{4}+\dfrac{1\cdot 3}{2!}\left(\dfrac{1}{4}\right)^{2}+\dfrac{1\cdot 3 \cdot 5}{3!} \left(\dfrac{1}{4}\right)^{3}+........$$ to $$\infty$$ is ?

A
$$\sqrt{2}$$

B
$$2$$

C
$$\dfrac{1}{\sqrt{2}}$$

D
$$\sqrt{3}$$

Solution
Question 9
1 / -0

For natural numbers m, n if $${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n} = 1 + {a_1}y + {a_2}{y^2} + .......$$ and $$a_1 = a_2 = 10$$, Then $$(m,n)$$ is

A
$$(10,45)$$

B
$$(35,20)$$

C
$$(45,35)$$

D
$$(35,45)$$

Solution
Question 10
1 / -0

The value of $$(21{C}_{1}-10{C}_{1})+(21{C}_{2}-10{C}_{2})+(3{C}_{1}-10{C}^{3})+(21{C}_{4}-10{C}^{4})+.....(21{C}_{10}-10{C}_{10})$$ is

A
$${2}^{20}-{2}^{10}$$

B
$${2}^{21}-{2}^{11}$$

C
$${2}^{21}-{2}^{10}$$

D
$${2}^{20}-{2}^{9}$$

Solution