Arithmetic Progressions Test

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Arithmetic Progressions Test - 14

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Question 1
1 / -0

Progressions with an equal common difference are known as:

A
Harmonic Progression

B
Arithmetic Progression

C
Geometric Progression

D
none of these

Solution

Progressions with an equal common difference are known as Arithmetic Progression. i.e the difference between any two consecutive terms is constant throughout the series, this constant difference is called common difference usually denoted by the letter d

if $a$ is the $1^{s t}$ term, d is a common difference, then the AP is represented by $a, a + d, a + 2 d, a + 3 d$ .......... $a + (n - 1) d$

Hence, the correct option is (B).
Question 2
1 / -0

If $a, 7, b, 23, c$ are in A.P. then the value of ‘ $c$ ’ is:

A
$8$

B
$31$

C
$- 1$

D
$0$

Solution

Let d be the common difference. Then

$a_{5} = c = a + 4 d$ ……….(i)

$a_{2} = a + d = 7$ ……….(ii)

$a_{4} = a + 3 d = 23$ ……….(iii)

Solving eq. (ii) and (iii),

we get $a = - 1$ and $d = 8$

$∴ c = a + 4 d = - 1 + 4 \times 8 = 31$

Hence, the correct option is (B).
Question 3
1 / -0

The next term of the A.P. $\sqrt{18}, \sqrt{32}$ and $\sqrt{50}$ is:

A
$\sqrt{72}$

B
$\sqrt{84}$

C
$\sqrt{80}$

D
$\sqrt{64}$

Solution

Given, $\sqrt{18}, \sqrt{32}, \sqrt{50}$

$\Rightarrow 3 \sqrt{2}, 4 \sqrt{2}, 5 \sqrt{2}$

$∴ d = 4 \sqrt{2} - 3 \sqrt{2} = \sqrt{2}$

Therefore, next term is $5 \sqrt{2} + \sqrt{2} = 6 \sqrt{2} = \sqrt{72}$

Hence, the correct option is (A).
Question 4
1 / -0

The next two terms of the AP : $k, 2 k + 1, 3 k + 2, 4 k + 3$ , ………… are:

A
$4 k + 4$ and $4 k + 5$

B
$5 k$ and $6 k$

C
$5 k + 4$ and $6 k + 5$

D
$5 k + 5$ and $6 k + 6$

Solution

Explanation:

Given: $k, 2 k + 1, 3 k + 2, 4 k + 3$ ,....

Here $d = 2 k + 1 - k = k + 1$

Therefore, the next two terms are:

$4 k + 3 + k + 1 = 5 k + 4$

and $5 k + 4 + k + 1 = 6 k + 5$

Hence, the correct option is (C).
Question 5
1 / -0

If $a, b$ and $c$ are in A. P., then the value of $\frac{a - b}{b - c}$ is:

A
$\frac{a}{b}$

B
$\frac{c}{a}$

C
$\frac{b}{c}$

D
$1$

Solution

If $a, b$ and $c$ are in A.P.,

$b - a = c - b$

$- (a - b) = - (b - c)$

$a - b = b - c$

dividing both sides by $b - c$

$\frac{a - b}{b - c} = \frac{b - c}{b - c}$

$\frac{a - b}{b - c} = 1$

Hence, the correct option is (D).
Question 6
1 / -0

The value of ‘ $k$ ’ for which the numbers $x, 2 x + k, 3 x + 6$ are in A.P. is:

A
4 $2$

B
$5$

C
$3$

D
$6$

Solution

$2 x + k - (x) = (3 x + 6) - (2 x + k)$

$2 x + k - x = 3 x + 6 - 2 x - k$

On simplifying the LHS and RHS, we will get,

$x + k = x + 6 - k$

On taking $x$ and $k$ from the LHS to the RHS we get,

$x + 6 - k - x - k = 0$

On simplifying further, we will get

$6 - 2 k = 0$

On taking $- 2 k$ from the LHS to RHS, we get

$6 = 2 k$

We can also write it as,

$2 k = 6$

Now, on dividing both the sides of the equation by $2$ , we get

$k = 3$

Hence, when the value of $k = 3$ , the numbers $x, 2 x + k$ and $3 x + 6$ form an AP. So, the AP will be $x, 2 x + 3, 3 x + 6$ .

Hence, the correct option is (C).
Question 7
1 / -0

The first four terms of the sequence $a_{n} = 2 n + 3$ are:

A
$5, 7, 9, 11$

B
$5, 8, 11, 14$

C
$1, 3, 5, 7$

D
$3, 5, 7, 9$

Solution

Given $a_{n} = 2 n + 3$

$∴ a_{1} = 2 \times 1 + 3 = 2 + 3 = 5$

$a_{2} = 2 \times 2 + 3 = 4 + 3 = 7$

$a_{3} = 2 \times 3 + 3 = 6 + 3 = 9$

$a_{4} = 2 \times 4 + 3 = 8 + 3 = 11$

Therefore, the first four terms are $5, 7, 9, 11$ .

Hence, the correct option is (A).
Question 8
1 / -0

If $a, b$ and $c$ are in A.P, then $b =$ ?

A
$\frac{a + c}{2}$

B
$\frac{c - a}{2}$

C
$\frac{a - c}{2}$

D
none of these

Solution

If $a, b$ and $c$ are in A.P., then

$b - a = c - b$

$2 b = a + c$

$b = \frac{a + c}{2}$

Hence, the correct option is (A).
Question 9
1 / -0

Which of the following is not an A.P.?

A
$a, 2 a, 3 a, 4 a$

B
$- 1.2, - 3.2, - 5.2, - 7.2$

C
$2, 4, 8, 16$

D
$2, \frac{5}{2}, 3, \frac{7}{2}$

Solution

In $2, 4, 8, 16$ , ……..

$d = a_{2} - a_{1} = 4 - 2 = 2$

And $d = a_{3} - a_{2} = 8 - 4 = 4$

Also $d = a_{4} - a_{3} = 16 - 8 = 8$

Here, common difference is not the same for all terms, therefore, it is not an AP.

Hence, the correct option is (C).
Question 10
1 / -0

Which of the following is an A.P.?

A
$1^{2}, 5^{2}, 7^{2}, 73$

B
$\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}$

C
$1, 3, 9, 27$

D
$1^{2}, 3^{2}, 5^{2}, 7^{2}$

Solution

In $1^{2}, 5^{2}, 7^{2}, 73, \dots \dots \dots . = 1, 25, 49, 73$ , ………

$d = a_{2} - a_{1} = 25 - 1 = 24$

And $d = a_{3} - a_{2} = 49 - 25 = 24$

Also $d = a_{4} - a_{3} = 73 - 49 = 24$

Here, common difference is same for all terms, therefore, it is an AP.

Hence, the correct option is (A).
Question 11
1 / -0

The common difference of the A.P whose $a_{n} = - 3 n + 7$ is:

A
$2$

B
$3$

C
$- 3$

D
$1$

Solution

Given: $a_{n} = - 3 n + 7$

Putting $n = 1, 2, 3$ , we get

$a = - 3 \times 1 + 7 = - 3 + 7 = 4$

$a_{2} = - 3 \times 2 + 7 = - 6 + 7 = 1$

$a_{3} = - 3 \times 3 + 7 = - 9 + 7 = - 2$

∴ Common difference $(d) = a_{2} - a = 1 - 4 = - 3$

Hence, the correct option is (C).
Question 12
1 / -0

If $a, b$ and $c$ are in A.P., then the relation between them is given by:

A
$2 b = a + c$

B
$2 c = a + b$

C
$2 a = b + c$

D
$a = b + c$

Solution

We know that if $a, b, c$ , .... are in AP then $2^{n d}$ term - $1^{s t}$ term $= 3^{r d}$ term $- 2^{n d}$ term

$b - a = c - b$

$\Rightarrow 2 b = a + c$

Hence, the correct option is (A).
Question 13
1 / -0

The list of numbers $- 10, - 6, - 2, 2,$ ……….. is:

A
an AP with $d = 8$

B
an AP with $d = - 4$

C
an AP with $d = 4$

D
not an AP

Solution

Given: list of numbers $- 10, - 6, - 2, 2$ , ………..

Here, $- 6 - (- 10) = - 6 + 10 = 4$

$- 2 - (- 6) = - 2 + 6 = 4$

$2 - (- 2) = 2 + 2 = 4$

Therefore, the list of given numbers is an A.P with a common difference of $4$ .

Hence, the correct option is (C).
Question 14
1 / -0

The common difference of the A.P. can be:

A
only zero

B
only negative

C
positive, negative or zero

D
only positive

Solution

The common difference of the A.P. can be positive, e.g. $1, 2, 3, 4$ ..... $d$ is $+ v e$ and series is increasing.

Negative e.g $4, 3, 2, 1$ ...... $d$ is $- v e$ and series are decreasing.

Or zero also and the AP becomes constant e.g $4, 4, 4, 4$ ......

Hence, the correct option (C).
Question 15
1 / -0

If the angles of a right-angled triangle are in A.P. then the angles of that triangle will be:

A
$45 °, 45 °, 90 °$

B
$20 °, 70 °, 90 °$

C
$40 °, 50 °, 90 °$

D
$30 °, 60 °, 90 °$

Solution

Let the three angles of a triangle be $a - d, a$ and $a + d$ .

$∴ a - d + a + a + d = 180 °$

$\Rightarrow 3 a = 180 °$

$\Rightarrow a = 60 °$

Therefore, one angle is $60 °$ and the other is $90 °$ (given).

Let the third angle be $x °$ , then

$60 ° + 90 ° + x ° = 180 °$

$\Rightarrow 150 ° + x ° = 0 °$

$\Rightarrow x ° = 180 ° - 150 ° = 30 °$

Therefore, the angles of the right-angled triangle are $30 °, 60 °, 90 °$ .

Hence, the correct option is (D).

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Arithmetic Progressions Test - 14

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Arithmetic Progressions Test - 14

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Question 1

Harmonic Progression

Arithmetic Progression

Geometric Progression

none of these

Solution

Question 2

8

31

-1

0

Solution

Question 3

72

84

80

64

Solution

Question 4

4k + 4 and 4k + 5

5k and 6k

5k + 4 and 6k + 5

5k + 5 and 6k + 6

Solution

Question 5

ab

ca

bc

1

Solution

Question 6

4 2

5

3

6

Solution

Question 7

5, 7, 9, 11

5, 8, 11, 14

1, 3, 5, 7

3, 5, 7, 9

Solution

Question 8

a+c2

c-a2

a-c2

none of these

Solution

Question 9

a, 2a, 3a, 4a

– 1.2, – 3.2, – 5.2, – 7.2

2, 4, 8, 16

2, 52, 3, 72

Solution

Question 10

12, 52, 72, 73

3, 6, 9, 12

1, 3, 9, 27

12, 32, 52, 72

Solution

Question 11

2

3

-3

1

Solution

Question 12

2b = a + c

2c = a + b

2a = b + c

a = b + c

Solution

$8$

$31$

$- 1$

$0$

$\sqrt{72}$

$\sqrt{84}$

$\sqrt{80}$

$\sqrt{64}$

$4 k + 4$ and $4 k + 5$

$5 k$ and $6 k$

$5 k + 4$ and $6 k + 5$

$5 k + 5$ and $6 k + 6$

$\frac{a}{b}$

$\frac{c}{a}$

$\frac{b}{c}$

$1$

4 $2$

$5$

$3$

$6$

$5, 7, 9, 11$

$5, 8, 11, 14$

$1, 3, 5, 7$

$3, 5, 7, 9$

$\frac{a + c}{2}$

$\frac{c - a}{2}$

$\frac{a - c}{2}$

$a, 2 a, 3 a, 4 a$

$- 1.2, - 3.2, - 5.2, - 7.2$

$2, 4, 8, 16$

$2, \frac{5}{2}, 3, \frac{7}{2}$

$1^{2}, 5^{2}, 7^{2}, 73$

$\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}$

$1, 3, 9, 27$

$1^{2}, 3^{2}, 5^{2}, 7^{2}$

$2$

$3$

$- 3$

$1$

$2 b = a + c$

$2 c = a + b$

$2 a = b + c$

$a = b + c$

an AP with $d = 8$

an AP with $d = - 4$

an AP with $d = 4$

$45 °, 45 °, 90 °$

$20 °, 70 °, 90 °$

$40 °, 50 °, 90 °$

$30 °, 60 °, 90 °$