Progressions - Aptitude MCQ Questions and Solutions with Explanations Page-7 section-1

61.
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

A. 5, 10, 15, 20

B. 4, 10, 16, 22

C. 3, 7, 11, 15

D. None of these

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62.
If S₁ is the sum of an arithmetic progression of ‘n’ odd number of terms and S₂ is the sum of the terms of the series in odd places, then $$\frac{{{S_1}}}{{{S_2}}}$$

A. $$\frac{{2n}}{{n + 1}}$$

B. $$\frac{n}{{n + 1}}$$

C. $$\frac{{n + 1}}{{2n}}$$

D. $$\frac{{n - 1}}{n}$$

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63.
If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is

A. 3200

B. 1600

C. 200

D. 2800

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64.
The n^th term of an A.P., the sum of whose n terms is S_n, is

A. S_n + S_{n - 1}

B. S_n - S_{n - 1}

C. S_n + S_{n + 1}

D. S_n - S_{n + 1}

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65.
The sum of first n odd natural numbers in

A. 2n - 1

B. 2n + 1

C. n²

D. n² - 1

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66.
The sum of n terms of an A.P. is 3n² + 5n, then 164 is its

A. 24^th term

B. 27^th term

C. 26^th term

D. 25^th term

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67.
The common difference of the A.P. $$\frac{1}{3},$$ $$\frac{{1 - 3b}}{3},$$ $$\frac{{1 - 6b}}{3},$$ . . . . . . is

A. $$\frac{1}{3}$$

B. $$ - \frac{1}{3}$$

C. $$ - b$$

D. $$b$$

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68.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of (p + q) terms will be

A. 0

B. p - q

C. p + q

D. -(p + q)

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Answer & Solution

Answer: Option D

Solution:

$$\eqalign{ & {\text{Sum}}\,{\text{of}}\,p\,{\text{terms}} = q \cr & i.e., \cr & {S_p} = \frac{p}{2}\left[ {2a + \left( {p - 1} \right)d} \right] = q \cr & \Rightarrow p\left[ {2a + \left( {p - 1} \right)d} \right] = 2q \cr & \Rightarrow 2ap + p\left( {p - 1} \right)d = 2q\,.....\left( 1 \right) \cr & {\text{and}}\,{\text{sum}}\,{\text{of}}\,q\,{\text{terms}} = p \cr & i.e., \cr & {S_q} = \frac{q}{2}\left[ {2a + \left( {q - 1} \right)d} \right] = p \cr & \Rightarrow q\left[ {2a + \left( {q - 1} \right)d} \right] = 2p \cr & \Rightarrow 2aq + q\left( {q - 1} \right)d = 2p\,.....\left( 2 \right) \cr & {\text{Subtracting}}\,\left( 2 \right)\,{\text{from}}\,\left( 1 \right) \cr} $$
$$ \Rightarrow 2a\left( {p - q} \right) + \left\{ {{p^2} - p - {q^2} + q} \right\}d$$ $$ = 2q - 2p$$
$$ \Rightarrow 2a\left( {p - q} \right) + \left\{ {{p^2} - {q^2} - \left( {p - q} \right)} \right\}d$$ $$ = - 2\left( {p - q} \right)$$
$$ \Rightarrow 2a\left( {p - q} \right) + \left\{ {\left( {p + q} \right)\left( {p - q} \right) - \left( {p - q} \right)} \right\}d$$ $$ = - 2\left( {p - q} \right)$$
$$ \Rightarrow 2a\left( {p - q} \right) + \left( {p - q} \right)\left[ {p + q - 1} \right]d$$ $$ = - 2\left( {p - q} \right)$$ $${\text{Dividing}}\,{\text{by}}$$ $$\,\left( {p - q} \right)$$
$$\eqalign{ & \Rightarrow 2a + \left( {p + q - 1} \right)d = - 2\,.....\left( 3 \right) \cr & {S_{p + q}} = \frac{{p + q}}{2}\left[ {2a + \left( {p + q - 1} \right)d} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{p + q}}{2}\left[ { - 2} \right]\,\,\,\left[ {{\text{From}}\,\left( 3 \right)} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = - \left( {p + q} \right) \cr} $$

69.
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is :

A. 13

B. 9

C. 21

D. 17

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70.
If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is

A. $$\frac{{ab}}{{2\left( {b - a} \right)}}$$

B. $$\frac{{ab}}{{b - a}}$$

C. $$\frac{{3ab}}{{2\left( {b - a} \right)}}$$

D. None of these

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Progressions - Aptitude MCQ Questions and Solutions with Explanations

61. If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

Answer & Solution

62. If S1 is the sum of an arithmetic progression of ‘n’ odd number of terms and S2 is the sum of the terms of the series in odd places, then $$\frac{{{S_1}}}{{{S_2}}}$$

Answer & Solution

63. If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is

Answer & Solution

64. The nth term of an A.P., the sum of whose n terms is Sn, is

Answer & Solution

65. The sum of first n odd natural numbers in

Answer & Solution

66. The sum of n terms of an A.P. is 3n2 + 5n, then 164 is its

Answer & Solution

67. The common difference of the A.P. $$\frac{1}{3},$$ $$\frac{{1 - 3b}}{3},$$ $$\frac{{1 - 6b}}{3},$$ . . . . . . is

Answer & Solution

68. If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of (p + q) terms will be

Answer & Solution

69. If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is :

Answer & Solution

70. If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is

Answer & Solution

61.
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

62.
If S₁ is the sum of an arithmetic progression of ‘n’ odd number of terms and S₂ is the sum of the terms of the series in odd places, then $$\frac{{{S_1}}}{{{S_2}}}$$

63.
If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 term is

64.
The n^th term of an A.P., the sum of whose n terms is S_n, is

65.
The sum of first n odd natural numbers in

66.
The sum of n terms of an A.P. is 3n² + 5n, then 164 is its

67.
The common difference of the A.P. $$\frac{1}{3},$$ $$\frac{{1 - 3b}}{3},$$ $$\frac{{1 - 6b}}{3},$$ . . . . . . is

68.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of (p + q) terms will be

69.
If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is :

70.
If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is