Arithmetic Progressions Test

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Arithmetic Progressions Test - 26

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Weekly Quiz Competition

Question 1
1 / -0

Find the common difference in the series: 0.2, 0.9, 1.6 ..............

A
0.5

B
0.1

C
0.7

D
0.6

Solution

First term, $$a_1 = 0.2$$

Second term, $$a_2 = 0.9$$

Common difference, $$d = a_2 - a_1$$

$$d= 0.9 - 0.2$$

$$\therefore d = 0.7$$
Question 2
1 / -0

Find the sum of the arithmetic series $$6 + 12 + 18 + ....... 96$$.

A
$$815$$

B
$$816$$

C
$$817$$

D
$$819$$

Solution

Given series is $$6+12+18+....96$$.
Here $$a = 6$$; last term, $$l = 96, d = 12-6 =6$$
We know that, last term, $$l = a + (n - 1) d$$
$$\Rightarrow 96 = 6 + (n - 1) 6$$
$$\Rightarrow 96 = 6 + 6n - 6$$
$$\Rightarrow 96 = 6n$$
$$\Rightarrow n = \dfrac{96}{6}$$
$$\Rightarrow n = 16$$ = total number of terms in the series

We know that,

Sum $$(S_n) = \dfrac{n}{2}$$ [First term $$+$$ Last term]

$$= \dfrac{16}{2} [6 + 96]$$

$$= 8 [102]$$

Therefore, $$S_{16} = 816$$
Question 3
1 / -0

Find the common difference, if the sum of first n terms will be (n - 2) (n - 1).

A
1

B
2

C
3

D
4

Solution

The sum of first n terms, $$S_n = \dfrac{n}{2} (2a + (n - 1)d)$$
On comparing the equations, we get
$$\dfrac{n}{2} (2a + (n - 1) d) = (n - 2)(n + 1)$$
$$(2a - d) n + dn^2 = n^2 - n - 2$$
On comparing the factors of $$n^2$$, we find the value of $$ d= 1$$
$$\therefore$$ The common difference is 1.
Question 4
1 / -0

Find the common difference in the sequence 4, 8, 12, 16, .......... 20

A
8

B
3

C
4

D
2

Solution

First term, $$a_1 = 4$$
Second term, $$a_2 = 8$$
Common difference, $$d = a_2 = a_1$$
$$d = 8 - 4 = 4$$
$$\therefore$$ The common difference is 4.
Question 5
1 / -0

_____ is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

A
Geometric value

B
Geometric series

C
Arithmetic progression

D
Arithmetic mean

Solution

$$\text {Arithmetic progression}$$ is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Ex: $$1, 2, 3, 4 $$ ..... In this sequence, the first number is $$1$$ and the next number is $$2$$. The difference between the two numbers is $$1$$ and so on.

By adding $$1$$ to the preceding number, we are forming the next term and so on, and the series is in $$AP$$.
Question 6
1 / -0

Find the number of terms in an A.P: $$-1,\ -5,\ -9,\ ..,\ - 197$$

A
48

B
49

C
50

D
47

Solution

Given: AP: $$-1, \ -5,\ -9,\ ...,\ -197$$

Here, First term $$a = -1$$

Common difference $$d = a_2 - a_1 = -5 - (-1) = - 4$$

And last term $$a_n = - 197$$

We know that, $$a_n = a + (n - 1) d$$

$$\Rightarrow -197 = -1 + (n - 1) - 4$$

$$\Rightarrow -197 = -1 - 4n + 4$$

$$\Rightarrow -197 + 1 - 4 = -4n$$

$$\Rightarrow n = 50$$

$$\therefore$$ There are 50 terms in the A.P
Question 7
1 / -0

Which term of the$$ A.P. 2, 9, 16, 23 .............$$ is $$100$$?

A
$$15$$

B
$$10$$

C
$$11$$

D
$$12$$

Solution

$$a = 2$$
common difference, $$d = a_2 - a_1, a - 2 = 7$$
we know that, $$a_n = a + (n - 1) d$$
$$a_n = 100$$
$$100 = a + (n - 1) 7$$
$$100 = 2 + 7n - 7$$
$$100 - 2 + 7 = 7n$$
$$\dfrac{105}{7} = n$$
$$n = 15$$
Hence, $$15th$$ term of $$A.P$$. is $$100$$.
Question 8
1 / -0

Find 11th term of the A.P. : -3, 1, 5, ........

A
36

B
37

C
38

D
39

Solution

First term, $$a_1 = -3$$
Common difference, $$d = a_2 - a_1 = 1 - (-3) = 4$$

We know that, $$a_n = a_1 + (n - 1) d$$
$$a_{11} = - 3 + (11- 1) 4$$
$$= -3 + (10) 4$$
$$= - 3 + 40$$
$$a_{11} = 37$$
Question 9
1 / -0

The sum of n terms of an arithmetic sequence can be calculated by .........................

A
$$S_n = \dfrac{n}{2} [2a+ (n - 1) d]$$

B
$$S_n = \dfrac{n}{2} [2a+ (n + 1) d]$$

C
$$S_n = \dfrac{2n}{2} [2a+ (n - 1) d]$$

D
$$S_n = \dfrac{2n}{2} [2a- (n - 1) d]$$

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
Question 10
1 / -0

Choose the correct statement(s):
$$A$$: Every sequence is a progression.
$$B$$: Every progression is a sequence.

A
Only $$A$$

B
Only $$B$$

C
Both $$A$$ and $$B$$

D
None

Solution

The difference between a progression and a sequence is that a progression has a specific rule to calculate its next term from its previous term, whereas a sequence can be based on a logical rule like 'a group of prime numbers'.
Thus, every progression is a sequence but every sequence is not a sequence.

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Arithmetic Progressions Test - 26

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Arithmetic Progressions Test - 26

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Question 1

0.5

0.1

0.7

0.6

Solution

Question 2

$$815$$

$$816$$

$$817$$

$$819$$

Solution

Question 3

1

2

3

4

Solution

Question 4

8

3

4

2

Solution

Question 5

Geometric value

Geometric series

Arithmetic progression

Arithmetic mean

Solution

Question 6

48

49

50

47

Solution

Question 7

$$15$$

$$10$$

$$11$$

$$12$$

Solution

Question 8

36

37

38

39

Solution

Question 9

$$S_n = \dfrac{n}{2} [2a+ (n - 1) d]$$

$$S_n = \dfrac{n}{2} [2a+ (n + 1) d]$$

$$S_n = \dfrac{2n}{2} [2a+ (n - 1) d]$$

$$S_n = \dfrac{2n}{2} [2a- (n - 1) d]$$

Solution

Question 10

Only $$A$$

Only $$B$$

Both $$A$$ and $$B$$

None

Solution

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