Sequences and Series Test

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Sequences and Series Test - 25

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle \frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+......+\frac{1}{35^{2}-1}=$$

A
$$\displaystyle \frac{4}{17}$$

B
$$\displaystyle \frac{17}{72}$$

C
$$\displaystyle \frac{19}{72}$$

D
$$\displaystyle \frac{17}{36}$$

Solution

given series $$=\sum _{n=1} ^{17} \dfrac{1}{(2n+1)^2 -1}$$

$$=\sum \dfrac{1}{4n^4+4n} =\sum \dfrac{1}{4n} \times \dfrac{1}{n+1}$$

solving this by integration, we will get
$$=\dfrac{17}{72}$$
Question 2
1 / -0

Find the sum of first three terms of the following sequence whose $$n^{th}$$ term is $$5n^2-2$$.

A
$$68$$

B
$$86$$

C
$$69$$

D
$$82$$

Solution

Given, $$ T_n = 5n^2 - 2 $$
So, $$ T_1= 5-2 = 3 $$
$$ T_2 = 5(2^2) -2 = 18 $$
$$ T_3 = 5(3^2) -2 = 47 $$

So, Sum of first three terms $$ = 3+18+47 = 68 $$
Question 3
1 / -0

Find the sum of 100 terms of the series $$1(3)+3(5)+5(7)+\cdots\cdots$$

A
1353300

B
1353400

C
1353200

D
1353100

Solution

Given
$$S_n=1(3)+3(5)+5(7)+......$$

It can be written as:
$$S_n=\displaystyle\sum^{100}_{n=1}(2n-1)(2n+1)$$

$$S_n=\displaystyle\sum^{100}_{n=1}(4n^2+2n-2n-1)$$

$$S_n=\displaystyle\sum^{100}_{n=1}(4n^2-1)$$

$$S_n=\displaystyle\sum^{100}_{n=1}4n^2-\displaystyle\sum^{100}_{n=1}1$$

$$S_n=\dfrac{4\times 100(101)(201)}{6}-100$$

$$S_n=1353400-100$$

$$S_n=1353300$$

Remember
$$\displaystyle\sum^{n}_{n=1}n^2=\dfrac{n(n+1)(2n+1)}{6}$$
$$\displaystyle\sum^{n}_{n=1}1=n$$
$$\displaystyle\sum^{n}_{n=1}n=\dfrac{n(n+1)}{2}$$.
Question 4
1 / -0

Find the sum of odd numbers between $$2$$ and $$76$$.

A
$$1,350$$

B
$$1,443$$

C
$$1,342$$

D
$$1,360$$

Solution

The sum of first n terms of arithmetic series formula can be written as,
$$S_n = \dfrac{n}{2} [2a + (n - 1)d]$$ .............. (1)
The list of odd numbers between $$2$$ and $$76$$ is $$ 3, 5, 7, 9, ........... 75$$
Where $$n =$$ number of terms $$= ?$$
$$a =$$ first odd number $$\Rightarrow$$ $$3$$
$$d = $$ common difference of A.P. $$\Rightarrow $$ $$2$$
$$T_n =$$ Last term $$\Rightarrow$$ $$75$$
$$T_n = a+ (n - 1) d$$
$$75 = 3 + (n - 1)2$$
$$ 75 = 3 + 2n - 2$$
$$75 - 3 + 2 = 2n$$
$$74 = 2n$$
$$n = 37$$
Apply the given data in the formula (1),
$$S_{37} = \dfrac{37}{2} [2 \times 3 + (37 - 1)2]$$
$$= 18.5 [6 + (36)2]$$
$$= 18.5 [6 + 72]$$
$$= 18.5 [78]$$
$$= 1,443$$
So, the sum of odd numbers between $$2$$ and $$76$$ is $$1,443$$.
Question 5
1 / -0

Find the next number of the series.
$$7, 10, 8, 11, 9, 12,....$$

A
9

B
10

C
8

D
7

Solution

The next number is $$10$$.
The first $$2$$ numbers difference is $$3$$.
The second $$2$$ numbers difference is $$2$$.
So, this is simple alternating addition of $$ 3$$ and subtraction of $$2$$.
Question 6
1 / -0

Which pair of numbers comes next?
$$25, 23, 5, 20, 16, 5,....$$

A
$$11, 5$$

B
$$5, 6$$

C
$$10, 5$$

D
$$9, 5$$

Solution

This is an alternating subtraction series with the interpolation of a random number, $$5$$, as every third number.
In the subtraction series, $$2$$ is subtracted, then $$3$$ then $$4$$, and so on.
The next pair of numbers are $$11$$ and $$5$$.
Question 7
1 / -0

Complete the pattern: $$3, 5, 8, 13,$$ ___

A
21

B
22

C
23

D
24

Solution

The next number is $$21$$.
We get the next number by adding the previous two numbers.
Question 8
1 / -0

Look at this series: $$32, 31, 33, 32, 34, 33,...$$. What is the next number?

A
34

B
32

C
35

D
36

Solution

The next number is $$35$$.
Here first $$1$$ is subtracted and then $$2$$ is added, and so on.
Question 9
1 / -0

In a certain sequence of 8 numbers, each number after the first is 1 more than the previous number .If the first number is -5, how many of the numbers in the sequence are positive?

A
None

B
One

C
Two

D
Three

E
Four

Solution
Question 10
1 / -0

Which pair of numbers comes next?
$$10, 18, 12, 21, 15 ,25, ,....$$

A
$$19, 30$$

B
$$19, 27$$

C
$$30, 19$$

D
$$27, 19$$

Solution

This is an alternating addition series
First series = $$10, 10+2=12, 12+3=15$$. $$\therefore$$ next number= $$15+4=19$$
Second series = $$18, 18+3=21, 21+4=25$$ $$\therefore$$ next number = $$25+5=30$$
The next pair of numbers are $$19, 30$$.

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Sequences and Series Test - 25

Result

Sequences and Series Test - 25

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Question 1

$$\displaystyle \frac{4}{17}$$

$$\displaystyle \frac{17}{72}$$

$$\displaystyle \frac{19}{72}$$

$$\displaystyle \frac{17}{36}$$

Solution

Question 2

$$68$$

$$86$$

$$69$$

$$82$$

Solution

Question 3

1353300

1353400

1353200

1353100

Solution

Question 4

$$1,350$$

$$1,443$$

$$1,342$$

$$1,360$$

Solution

Question 5

9

10

8

7

Solution

Question 6

$$11, 5$$

$$5, 6$$

$$10, 5$$

$$9, 5$$

Solution

Question 7

21

22

23

24

Solution

Question 8

34

32

35

36

Solution

Question 9

None

One

Two

Three

Four

Solution

Question 10

$$19, 30$$

$$19, 27$$

$$30, 19$$

$$27, 19$$

Solution

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