Arithmetic Progressions Test

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Arithmetic Progressions Test - 51

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Question 1
1 / -0

In a zoo, there are $$210$$ visitors on first day, $$250$$ visitors on second day and $$290$$ visitors on third day, and so on in an arithmetic sequence. What is the total number of visitors on $$15^{th}$$ day?

A
$$7,960$$

B
$$7,250$$

C
$$7,660$$

D
$$7,350$$

Solution

The sequence is $$210, 250, 290 ..........$$
$$a = 210, d = 40$$
First find the common difference:
$$a_n = a_1 + (n - 1)d$$
$$a_{15} = 210 + (15 - 1) 40$$
$$= 210 + 14 \times 40$$
$$a_{15} = 770$$
We know that, $$S_n = \dfrac{n}{2} $$ [First term $$+$$ Last term]
$$= \dfrac{15}{2} [210 + 770]$$
$$= 7.5 [980]$$
$$S_{15} = 7,350$$
So, the total number of visitors on $$15^{th}$$ day is $$7,350$$.
Question 2
1 / -0

There are $$200$$ students in IT department, $$210$$ students in Computer science department and $$220$$ students in Mechanic department in the college, and so on in an arithmetic sequence. What is the total number of students in $$8$$ departments?

A
$$1,880$$

B
$$1,760$$

C
$$1,660$$

D
$$1,560$$

Solution

The sequence is $$200, 210, 220 .....$$
$$a = 200, d = 10$$
First find the common difference:
$$a_n = a_1 + (n - 1 )d$$
$$a_8 = 200 + (8 - 1) 10$$
$$= 200 + 7 \times 10$$
$$a_8 = 270$$
We know that, $$S_n = \dfrac{n}{2} $$ [First term $$+$$ Last term]
$$= \dfrac{8}{2} [200 + 270]$$
$$= 4 [470]$$
$$S_8 = 1,880$$
So, the total number of students in $$8$$ departments is $$1,880$$.
Question 3
1 / -0

There are $$100$$ students in science department, $$150$$ students in computer science department in a higher secondary school and so on in an arithmetic sequence. What is the total number of students in $$10$$ departments?

A
$$3,960$$

B
$$3,250$$

C
$$3,660$$

D
$$3,560$$

Solution

The sequence is $$100, 150 .....$$
$$a = 100, d= 50$$
First find the common difference:
$$a_n = a_1 + (n - 1) d$$
$$a_{10} = 100 + (10 - 1) 50$$
$$= 100 + 9 \times 50$$
$$a_{10} = 550$$
We know that, $$S_n = \dfrac{n}{2} $$ [First term $$+$$ Last term]
$$= \dfrac{10}{2} [100 + 550]$$
$$= 5 [650]$$
$$S_{10} = 3,250$$
So, the total number of students in $$10$$ departments is $$3,250$$.
Question 4
1 / -0

In a national treasure library, there are 10 visitors on first day, 50 visitors on second day and 90 visitors on third day, and so on in an arithmetic sequence. What is the total number of visitors on 5th day?

A
460

B
450

C
440

D
430

Solution

The sequence is 10, 50, 90 ......
$$a = 10, d = 40$$
First find the common difference:
$$a_n = a_1 + (n - 1) d$$
$$a_5 = 10 + (5 - 1) 40$$
$$= 10 + 4 \times 40$$
$$a_5 = 170$$
We know that, $$S_n = \dfrac{n}{2}$$ [First term + Last term]
$$= \dfrac{5}{2} [10 + 170]$$
$$= 2.5 [180]$$
$$S_5 = 450$$
So, the total number of visitors on 5th day is 450
Question 5
1 / -0

The first term of an A.P. is $$2$$ and ninth term is $$58$$. What is the common difference?

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$

Solution

We know the formula, $$a_n = a + (n - 1) d$$
Given, $$a = 2; a_9 = 58$$
Therefore, $$a_9 = a + (9 -1 )d$$
$$\Rightarrow 58 = 2 + (8) d$$
$$\Rightarrow 58 - 2 = 8d$$
$$\Rightarrow 56 = 8d$$
$$\Rightarrow d = 7$$
Question 6
1 / -0

What is the sum of the first forty terms of the arithmetic sequence 120, 115, 110, 105....?

A
900

B
700

C
500

D
300

Solution

The sum of $$n$$ terms of an arithmetic sequence can be calculated by $$S_n = \dfrac{n}{2} [2a+ (n - 1) d]$$
Where,$$n$$ denotes the total number of terms
$$a =$$ first term
$$d =$$ common difference
$$S_n =$$ Sum of $$n$$ terms
$$\therefore$$ First Term: $$a = 120$$
Common Difference: $$d = -5$$
Total terms: $$n = 40$$

Sum of $$40$$ terms:
$$S = \dfrac{40}2\times(2\times120 + (40-1)\times (-5))$$

$$S = 20\times (240 - 39\times 5)$$
$$S = 900$$
Question 7
1 / -0

What is the sum of the first nineteenth terms of the arithmetic sequence 2, 6, 10, 14.....?

A
$$522$$

B
$$622$$

C
$$722$$

D
$$822$$

Solution

The sum of n terms of an arithmetic sequence can be calculated by $$S_n = \dfrac{n}{2} [2a+ (n - 1) d]$$
Where, n denotes the total number of terms
$$a $$= first term
$$d = $$common difference
$$S_n$$ = Sum of n terms
$$\therefore a = 2
d = 4
n = 19$$
$$S = \dfrac{n}{2}[2\times a+(n - 1)d]$$
$$S = \dfrac{19}{2}[2\times 2+(19 - 1)4]$$
$$S = \dfrac{19}{2}[4+72]$$
$$S = 9.5[76]$$
$$S =722$$
Question 8
1 / -0

If the $$6^{th}$$ term and $$12^{th}$$ term of an A.P. are $$30$$ and $$75$$ respectively, find the sum of its $$20$$ terms.

A
$$1275$$

B
$$1150$$

C
$$1140$$

D
$$1130$$

Solution

The $$nth$$ term and sum of n terms of an arithmetic sequence can be calculated by $$a_n = a+(n-1)d$$ & $$S_n = \dfrac{n}{2} [2a+ (n - 1) d]$$
Where, n denotes the total number of terms
a = first term
d = common difference
$$S_n$$ = Sum of n terms
$$\therefore$$ $$a+5d = 30$$ ($$6^{th}$$ term)
and $$a+11d = 75$$ ($$11^{th}$$ term)
Solving both, we get
$$d=7.5$$
$$a=-7.5$$
The sum of $$20$$ terms:
$$=\dfrac{20}{2}(2(-7.5)+(20-1)7.5)$$
$$= 10(17 \times 7.5)$$
$$=1275$$
Question 9
1 / -0

Which of the term of the A.P. 11, 22, 33, 44.... is 550?

A
$$20$$

B
$$30$$

C
$$40$$

D
$$50$$

Solution

Let the general form of A.P., $$a_n = a + (n-1)d$$
Common difference $$d = 11$$
and $$a_{n}=550$$
So,
$$a_n = a + (n-1)d$$
$$550 = 11 + (n-1)\times 11$$
$$550-11 = 11n-11$$
$$539+11= 11n$$
$$550 = 11n$$
$$n = 50$$
Question 10
1 / -0

In an arithmetic series, $$a_1 = 7$$ and $$a_{12}=29$$. Find the sum of the first 12 terms.

A
$$116$$

B
$$216$$

C
$$316$$

D
$$416$$

Solution

Given:
First term: $$a = 7$$
Last term, $$l = 29$$

Here, $$a_{n}=29$$

and $$n = 12$$

We know that $$S_{n}=\dfrac{n}{2}[a+l]$$

$$\therefore \ S_{12}=\dfrac{12}{2}[7+29]$$

$$=6\times 36$$

$$=216$$

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Re-Attempt Weekly Quiz Competition

Arithmetic Progressions Test - 51

Result

Arithmetic Progressions Test - 51

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Rank

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SHARING IS CARING

Question 1

$$7,960$$

$$7,250$$

$$7,660$$

$$7,350$$

Solution

Question 2

$$1,880$$

$$1,760$$

$$1,660$$

$$1,560$$

Solution

Question 3

$$3,960$$

$$3,250$$

$$3,660$$

$$3,560$$

Solution

Question 4

460

450

440

430

Solution

Question 5

$$5$$

$$6$$

$$7$$

$$8$$

Solution

Question 6

900

700

500

300

Solution

Question 7

$$522$$

$$622$$

$$722$$

$$822$$

Solution

Question 8

$$1275$$

$$1150$$

$$1140$$

$$1130$$

Solution

Question 9

$$20$$

$$30$$

$$40$$

$$50$$

Solution

Question 10

$$116$$

$$216$$

$$316$$

$$416$$

Solution

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