Sequences and Series Test

Result

Sequences and Series Test - 65

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Weekly Quiz Competition

Question 1
1 / -0

Sum of the series $$S={ 1 }^{ 2 }-{ 2 }^{ 2 }+{ 3 }^{ 2 }-{ 4 }^{ 2 }+......{ 2008 }^{ 2 }+{ 2009 }^{ 2 }$$.

A
2019045

B
1005004

C
2000506

D
None of these.

Solution
Question 2
1 / -0

If in a progression $${a_1},{a_2},{a_3},...,etc.,\left( {{a_r} - {a_{r + 1}}} \right)$$ bears a constant ratio with $${a_r}.{a_{r + 1}}$$ then the term of the progression are in

A
AP

B
GP

C
HP

D
none of there

Solution
Question 3
1 / -0

Suppose n be integer than 1, let $$a_n=\frac 1{\log _n2002}$$. Suppose $$b=a_2+a_3+a_4+a_5$$ and $$c=a_{10}+a_{11}+a_{13}+a_{14}$$, Then (b-c) equals

A
$$\frac 1{1001}$$

B
$$\frac 1{1002}$$

C
$$-1$$

D
$$-2$$

Solution
Question 4
1 / -0

If $$ (20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17} $$ $$ +\ldots+20(21)^{19}=k(20)^{19} $$ then $$ k $$ is equal to

A
$$400$$

B
$$100$$

C
$$441$$

D
$$420$$

Solution
Question 5
1 / -0

If $$\frac { 48 }{ (2)(3) } +\frac { 47 }{ (3)(4) } +\frac { 46 }{ (4)(5) } +........+\frac { 2 }{ (48)(49) } +\frac { 1 }{ (49)(50) } =\frac { 51 }{ 2 } +k(1+\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +.....+\frac { 1 }{ 50 } )$$, then K equals

A
$$-1$$

B
$$-\frac { 1 }{ 2 } $$

C
$$1$$

D
$$2$$

Solution
Question 6
1 / -0

$${ C }^{ 2n }{ C }_{ n }-{ { C }_{ 1 } }^{ 2n-2 }{ C }_{ n }+{ { C }_{ 2 } }^{ 2n-4 }{ C }_{ n }...$$ equals to

A
$${ 2 }^{ n }$$

B
$${ 2 }^{ n }(n+1)$$

C
$${ 2 }^{ n-1 }$$

D
$$\dfrac { (n+1){ 2 }^{ n-1 } }{ 2 } $$

Solution
Question 7
1 / -0

$${\log _5}2,\,\,{\log _6}\,2,\,\,{\log _{12}}\,\,2\,\,$$ are in

A
A.P

B
G.P

C
H.P

D
none of these

Solution
Question 8
1 / -0

$$\text{Find the value of } \log \sin 1^{\circ} . \log \sin 2^{\circ} \ldots \ldots \log \sin 179^{\circ} $$

A
1

B
0

C
-1

D
2

Solution

$$\textbf{Step-1: Apply standard angle of trigonometry function to get the required unknown.}$$

$$\text{We have, }$$

$$\log \sin 1^{\circ} . \log \sin 2^{\circ} \ldots \ldots \log \sin 179^{\circ} $$

$$\text{Above expression can be rewritten as}$$

$$\log \sin 1^{\circ} . \log \sin 2^{\circ} \ldots \log\sin 90^{\circ}\ \ldots \log \sin 179^{\circ} $$

$$\Rightarrow$$ $$0$$ $$\textbf{[As log 1 = 0 and sin}$$ $$\boldsymbol{90^{\circ} = 0]}$$

$$\textbf{Hence, option- B is correct answer}$$
Question 9
1 / -0

The value of 1+$${ i }^{ 1 }+{ i }^{ 4 }+{ i }^{ 5 }+.......{ i }^{ 2n }$$ is :

A
Positive

B
negative

C
zero

D
cannot be determined

Solution
Question 10
1 / -0

Find the A.M. of the series $$1,2,4,8,16 , \ldots , 2 ^ { n }$$

A
$$\frac { 2 ^ { n + 1 } - 1 } { n + 1 }$$

B
$$\frac { 2 ^ { n + 2 } - 1 } { n }$$

C
$$\frac { 2 ^ { n } - 1 } { n + 1 }$$

D
$$\frac { 2 ^ { n } - 1 } { n }$$

Solution