Arithmetic Progressions Test

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Arithmetic Progressions Test - 50

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Question 1
1 / -0

The arithmetic sequence is given as:- -7, -4, -1, 2, 5 .......... Find their common difference and next term of the sequence.

A
Common difference $$= 3$$; $$a_6 = 8$$

B
Common difference $$= 2$$; $$a_4 = 7$$

C
Common difference $$= 1$$; $$a_6 = 5$$

D
Common difference $$= 0$$; $$a_6 = 0$$

Solution

$$a_1 = - 7$$
$$a_2 = - 4$$
Common difference, $$d = a_2 - a_1$$
$$= -4 - (- 7)$$
$$= -4 + 7 = 3$$
$$d = 3$$
The next term will be $$a_6$$ So, $$a_5 + d = a_6$$
$$5 + 3 = 8$$
$$\therefore$$ The next term is 8
Question 2
1 / -0

If $$a_5 = 12$$ and $$a_{15} = 400$$. Find $$S_{10}$$

A
314

B
320

C
316

D
318

Solution

Using the formula,
$$a_n = a + (n - 1) d$$
So, $$a_5 = a + (5 - 1) d$$
$$12 = a + 4d $$ ............. (1)
$$a_{15} = a + (15 - 1) d$$
$$400 = a + 14 d $$ .......... (2)
Solving equation (1) & (2)
$$12 = a + 4 d$$
$$400 = a + 14 d$$
(-) (-) (-)
---------------------
$$- 388 = -10d$$
$$d= 38.8$$
Put the value of d in equation (1)
$$12 = a + 4 (38.8)$$
$$12 = a + 155. 2$$
$$a = - 143.2$$
The sum of first n term is given by $$S_n = \dfrac{n}{2} (2a + (n - 1)d)$$
$$S_{10} = \dfrac{10}{2} (2 (-143. 2) + (10 - 1) 38.8)$$
$$= 5 (- 286.4 + 349.2)$$
$$= 5 (62.8)$$
$$S_{10} = 314$$
Question 3
1 / -0

In an arithmetic sequence $$1, -2, -5, -8 $$ ....., then find the $$12^{th}$$ term.

A
$$-30$$

B
$$-32$$

C
$$33$$

D
$$34$$

Solution

$$12^{th}$$ term of the A.P. is $$a_{12} = a + (n - 1) d$$
Given series is $$1,-2,-5,-8...$$
Here $$a = 1, d = - 3, a_{12} = ?$$
Since $$a_{n} = a + (n-1)(d)$$
Therefore, $$a_{12} = 1 + (12- 1) (- 3)$$
$$ a_{12} = 1 + 11 \times (-3)$$
$$ a_{12} = 1 - 33$$
$$\Rightarrow a_{12} = - 32$$
Question 4
1 / -0

Find the common difference in the sequence: $$7, 14, 21, ............ 70$$

A
$$4$$

B
$$5$$

C
$$6$$

D
$$7$$

Solution

First term, $$a_1 = 7$$
Second term, $$a_2 = 14$$
Common difference, $$d = a_2 - a_1$$
$$= 14 - 7 \Rightarrow 7$$
$$\therefore$$ The common difference is 7.
Question 5
1 / -0

What is the common difference, if $$a_1 = 100$$ and $$a_2 = 250$$

A
$$100$$

B
$$150$$

C
$$200$$

D
$$210$$

Solution

Common difference, $$d = a_2 - a_1$$
$$= 250 - 100$$
$$d = 150$$
Question 6
1 / -0

A bus can travel $$10$$ m in the first minute, $$25$$ m the next minute, $$40$$ m the third minute and so on in an arithmetic sequence. What is the total distance the bus travels in $$10$$ minutes?

A
$$2,000$$

B
$$775$$

C
$$4,000$$

D
$$5,000$$

Solution

The sequence is 10, 25, 40 .....
$$a = 10, d = 15$$
First find the common difference:
$$a_n = a_1 + (n - 1) d$$
$$a_{10} = 10 + (10 - 1) 15$$
$$= 10 + 9 \times 15$$
$$a_{10} = 145$$
We know that, $$S_n = \dfrac{n}{2}$$ [First term + Last term]
$$= \dfrac{10}{2} [10 + 145]$$
$$= 5 [155]$$
$$S_{10} = 775$$
Hence, the bus will travel 775 m in 10 minutes.
Question 7
1 / -0

10 people live on the 1st floor of the apartment, 22 people on the second floor and 34 people on the third floor, and so on in an arithmetic sequence up to the 10th floor. What is the total number of people living in the apartment?

A
656

B
564

C
640

D
589

Solution

The sequence is 10, 22, 34 ......
$$a = 10, d = 22-10 = 12$$
$$a_n = a_1 + (n - 1) d$$
$$a_{10} = 10 + (10 - 1) 12$$
$$= 10 + 9 \times 12$$
$$a_{10} = 118$$

We know that, $$S_n = \dfrac{n}{2} $$ [First term + Last term]

$$= \dfrac{10}{2} [10 + 118]$$

$$= 5 [128]$$

$$S_{10} = 640$$

Hence, the total number of people living in the apartment is $$640$$.
Question 8
1 / -0

There are 25 tourist in the first travel, 50 tourist the second travel and 75 tourist the third travel in the museum and so on in an arithmetic sequence. Find the total number of tourist in 25 travels?

A
8,460

B
8,450

C
8250

D
8,125

Solution

The sequence is 25, 50, 75
$$a = 25, d = 25$$
First find the common difference:
$$a_n = a_1 + (n - 1) d$$
$$a_{25} = 25 + (25 - 1) 25$$
$$= 25 + 24 \times 25$$
$$a_{25} = 625$$
We know that, $$S_n = \dfrac{n}{2}$$ [First term + Last term]
$$= \dfrac{25}{2} [25 + 625]$$
$$= 12.5 [650]$$
$$S_{25} = 8,125$$
So, the total number of tourist in 25 travels is 8,125
Question 9
1 / -0

What is the $$21^{st}$$ term of the arithmetic sequence $$ -5, -2, 1, 4, 7 ......?$$

A
$$51$$

B
$$55$$

C
$$54$$

D
$$53$$

Solution

Given sequence is $$-5, -2, 1, 4, 7 ......... $$
To find out: The $$21^{st}$$ term of the sequence.

We know that, the $$n^{th}$$ term of an A.P. is $$t_n=a+(n-1)d$$
Here $$a = - 5,\ n=21\ and\ d = 3$$

$$\therefore \ t_{21} = - 5 + (21 - 1) 3$$
$$\Rightarrow - 5 + (20 ) 3$$
$$\Rightarrow - 5 + 60$$
$$\therefore \ t_{21} = 55$$

Hence, the $$21^{st}$$ term of the given sequence is $$55$$.
Question 10
1 / -0

Find the difference of the arithmetic progression an if $$a_1 = 7$$ and $$a_3 = 16$$

A
$$4.5$$

B
$$5$$

C
$$5.5$$

D
$$6$$

Solution

Given, $$a_1 = 7; a_3 = 16$$
Common difference, $$d = ?$$
$$a_3 = a_2 + d = a_1+2d$$
So, $$d = \dfrac{a_3 - a_1}{2}$$
$$= \dfrac{16 - 7}{2}$$
$$d = 4.5$$
Hence, the common difference is $$4.5$$.

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Re-Attempt Weekly Quiz Competition

Arithmetic Progressions Test - 50

Result

Arithmetic Progressions Test - 50

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SHARING IS CARING

Question 1

Common difference $$= 3$$; $$a_6 = 8$$

Common difference $$= 2$$; $$a_4 = 7$$

Common difference $$= 1$$; $$a_6 = 5$$

Common difference $$= 0$$; $$a_6 = 0$$

Solution

Question 2

314

320

316

318

Solution

Question 3

$$-30$$

$$-32$$

$$33$$

$$34$$

Solution

Question 4

$$4$$

$$5$$

$$6$$

$$7$$

Solution

Question 5

$$100$$

$$150$$

$$200$$

$$210$$

Solution

Question 6

$$2,000$$

$$775$$

$$4,000$$

$$5,000$$

Solution

Question 7

656

564

640

589

Solution

Question 8

8,460

8,450

8250

8,125

Solution

Question 9

$$51$$

$$55$$

$$54$$

$$53$$

Solution

Question 10

$$4.5$$

$$5$$

$$5.5$$

$$6$$

Solution

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