Sequences and Series Test

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Sequences and Series Test - 38

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Weekly Quiz Competition

Question 1
1 / -0

Let $$S_n=\dfrac{1}{1+2008n}+\dfrac{1}{2+2008n}+...+\dfrac{1}{2009n}$$. Then $$\displaystyle\lim_{n\rightarrow\infty}S_n$$ equals?

A
$$log\left(\dfrac{1}{2008}\right)$$

B
$$log\left(1+\dfrac{1}{2008}\right)$$

C
$$log\left(\dfrac{1}{2009}\right)$$

D
$$log\left(1+\dfrac{1}{2009}\right)$$

Solution
Question 2
1 / -0

If $$\frac{a_2a_3}{a_1a_4}=\frac{a_2+a_3}{a_1+a_4}=3\left(\frac{a_2-a_3}{a_1-a_4}\right)$$ then $$a_1, a_2$$ and $$a_3$$ are in which progression?

A
A.P.

B
G.P.

C
H.P.

D
None of these

Solution
Question 3
1 / -0

The sum of all $$2$$-digit numbers divisible by $$5$$ is _________?

A
$$1035$$

B
$$1245$$

C
$$1230$$

D
$$945$$

Solution

The series will be : 10, 15, 20,.....,90,95
$$S_{n} = 10 + 15 + 20 +...... + 90 + 95 $$
here a will be first term
a = 10
& d = difference b/w any
two terms
d = 5
series containing n terms
& n = $$ 1 + \dfrac{\left ( Last\, term - first \,term \right ) } {d} $$
$$ = 1 + \dfrac{\left ( 95 - 10 \right )}{5} $$
n = 1 + 17 = 18
now, $$ S_{n}= \dfrac{n}{2} \left [ 2a + \left ( n - 1 \right )d \right ] $$
$$ = \dfrac{18}{2}\left [ 2\left ( 10 \right )+\left ( 18 - 1 \right )\left ( 5 \right ) \right ] $$

$$ = 9\left ( 20 + 85 \right ) $$
$$ S_{n} = 9\left ( 105 \right ) $$
$$ \boxed { Sn = 945} $$
Question 4
1 / -0

The $$15^{th}$$ terms of the series $$2\dfrac{1}{2} + 1\dfrac{7}{13} + 1\dfrac{1}{9} + \dfrac{20}{23} +$$_____ is

A
$$\dfrac{5}{12}$$

B
$$\dfrac{10}{21}$$

C
$$\dfrac{10}{39}$$

D
None of these
Solution
- On solving these improper fractions into proper fraction and by making numerators be 20.
we get series as ,
=> $$\dfrac{20}{8}$$+$$\dfrac{20}{13}$$+$$\dfrac{20}{18}$$+$$\dfrac{20}{23}$$+$$\dfrac{20}{28}$$+............

Therefore , here denominators are in AP having common difference be $$5$$.
So , $$15th$$ term of denominator be = $$8+14\times 5=78$$

Hence $$15th$$ term of series will be,

=> $$\dfrac{20}{78}$$ = $$\dfrac{10}{39}$$
Question 5
1 / -0

If $$S_r=\left |
\begin{array}{111}
2r & x & n(n+1)\\
6r^2-1 & y & n^2(2n+3)\\
4r^3-2nr & z & n^3(n+1)\\
\end{array}
\right |
$$ then $$\sum_\limits{r=1}^n S_r$$
does not depends on

A
x

B
y

C
n

D
all of these

Solution
Question 6
1 / -0

Insert the missing number in the given series : $$ 0,4,18,48, ?, 180$$

A
$$58$$

B
$$68$$

C
$$84$$

D
$$100$$

Solution

$$0=0\times1^2$$
$$4=1\times2^2$$
$$18=2\times3^2$$
$$48=3\times4^2$$
$$\therefore4\times5^2=4\times25=100$$ is the next term.
Question 7
1 / -0

Ten students of the physics department decided to go on a educational trip.They hired a mini bus for the trip, but the bus can only carry eight students at a time and each student goes at least once. Find the minimum number of trips the bus has to make so that each students can go for equal number of trips.

A
$$5$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

Ten students $$\longrightarrow$$ Eight students in bus $$2$$ students are excluded/not taken on a trip so we have exclude every students once, so that no. of trips made by each students are equal so no. of trips $$= \dfrac{10}{2} =5$$
$$A$$ is correct.
Question 8
1 / -0

For a sequence $$\left\{ { a }_{ n } \right\} ,{ a }_{ 1 }=2$$ and $$\dfrac { { a }_{ n+1 } }{ { a }_{ n } } =\dfrac { 1 }{ 3 }$$. Then $$\sum _{ r=1 }^{ 20 }{ { a }_{ r } } $$ is

A
$$\dfrac {20}{2}[4+19\times 3]$$

B
$$3(1-\dfrac {1}{3^{20}})$$

C
$$2(1-3^{20})$$

D
$$(1-\dfrac {1}{3^{20}})$$

Solution

Given that,

$$a_1=2$$, $$r=\dfrac{a_{n+1}}{a_n}=\dfrac{1}{3}$$

$$ \sum\nolimits_{r=1}^{20}{{{a}_{r}}=\dfrac{a\left( 1-{{r}^{n}} \right)}{1-r}} $$

$$ =\dfrac{2\left[ 1-{{\left( \dfrac{1}{3} \right)}^{20}} \right]}{1-\dfrac{1}{3}} $$ $$[n=20]$$

$$ =\dfrac{2\left[ 1-{{\left( \dfrac{1}{3} \right)}^{20}} \right]}{\dfrac{2}{3}} $$

$$ =3\left[ 1-{{\left( \dfrac{1}{3} \right)}^{20}} \right]$$

Hence this is the correct answer.
Question 9
1 / -0

Number of rectangles in the grid shown which are not squares is?

A
$$160$$

B
$$162$$

C
$$170$$

D
$$185$$

Solution
Question 10
1 / -0

$$1,2,1,4,3,8,9,5,27,16,?,?,?$$

A
$$7,36,64$$

B
$$7,25,64$$

C
$$9,25,64$$

D
$$7,25,49$$

Solution