Progressions (A.P., G.P., H.P.) Questions and Answers

Q: 1 + 3 + 5 + ........ 33 + 35 =

The sum is 324.The series adds consecutive odd numbers from 1 to 35. First count the terms, then use the fact that the sum of the first n odd numbers is n squared.\[ \begin{aligned} n &= \frac{35 + 1}{2} = 18 \\ S &= n^{2} = 18^{2} = 324 \end{aligned} \]The same result comes from the arithmetic series formula, number of terms times the average of the first and last terms:\[ S = \frac{18}{2}(1 + 35) = 9 \times 36 = 324 \]The other options are not the square of the number of terms; 423 simply reverses the digits of the correct answer.

Q: A clock strikes once at 1 O'clock, twice at 2 O'clock, thrice at 3 O'clock and so on. How many times in total will it strike in 24 hours ?

The clock strikes 156 times in 24 hours.In each 12-hour cycle it strikes 1 + 2 + ... + 12 times, and a day has two such cycles.\[ \begin{aligned} 1 + 2 + \cdots + 12 &= \frac{12 \times 13}{2} = 78 \\ 2 \times 78 &= 156 \end{aligned} \]144 is \(12 \times 12\), which treats every hour as 12 strokes, and 164 and 168 match no counting rule. The key step is remembering that the 1-to-12 cycle happens twice a day.

Q: The sum of all odd numbers from 1 to 51 is

The sum is 676.The odd numbers from 1 to 51 are 1, 3, 5, ..., 51. There are \(\frac{51 + 1}{2} = 26\) of them, and the sum of the first \(n\) odd numbers is \(n^{2}\):\[ 1 + 3 + 5 + \cdots + 51 = 26^{2} = 676 \]625 is \(25^{2}\), the sum up to 49, which leaves out 51. 576 is \(24^{2}\), and 696 does not come from the formula.The formula works because adding consecutive odd numbers builds perfect squares: 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and so on.

Q: How many terms are there in the series 201, 208, 215, ..... 369 ?

The series has 25 terms.201, 208, 215, ... is an arithmetic progression with first term 201 and common difference 7. If 369 is the \(n\)th term:\[ \begin{aligned} T_n &= a + (n-1)d \\ 369 &= 201 + (n-1) \times 7 \\ n - 1 &= 168 \div 7 = 24 \\ n &= 25 \end{aligned} \]24 is the number of gaps between terms, the answer you get by forgetting to add 1 for the first term. 26 and 23 are off-by-one slips in the other direction.

Important Instructions

81.

In an A.P, \(t_{7} - t_{4} = 20\). Which one of the following is false?

A.We cannot determine first term
B.Difference between first term and 10th term is 60
C.A.M. of 12th term and 20th term is 16th term
D.A.M. of 5th term and 30th term is 17th term

Exam Year(s)

TNPSC Group 1 - 2025, TNPSC - 2025,

Answer: D

Since \(t_{7}-t_{4}=3d=20\), the common difference is \(d=\tfrac{20}{3}\), which fixes d but not the first term, so (A) is true.

The difference between the 10th and 1st terms is 9d = 60, so (B) is true, and the A.M. of the 12th and 20th terms is the 16th term, so (C) is true. The A.M. of the 5th and 30th terms is the \(\tfrac{5+30}{2}=17.5\)th term, not the 17th, so (D) is false.

Since d = 20/3, the 10th term minus the first is 9 x 20/3 = 60, which agrees with statement (B).

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82.

1 + 3 + 5 + ........ 33 + 35 =

A.324
B.423
C.420
D.438

Exam Year(s)

Tripura Public Service Commission - 2015,

Answer: A

The sum is 324.

The series adds consecutive odd numbers from 1 to 35. First count the terms, then use the fact that the sum of the first n odd numbers is n squared.

\[ \begin{aligned} n &= \frac{35 + 1}{2} = 18 \\ S &= n^{2} = 18^{2} = 324 \end{aligned} \]

The same result comes from the arithmetic series formula, number of terms times the average of the first and last terms:

\[ S = \frac{18}{2}(1 + 35) = 9 \times 36 = 324 \]

The other options are not the square of the number of terms; 423 simply reverses the digits of the correct answer.

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83.

A clock strikes once at 1 O'clock, twice at 2 O'clock, thrice at 3 O'clock and so on. How many times in total will it strike in 24 hours ?

A.156
B.144
C.164
D.168

Exam Year(s)

Tripura Public Service Commission - 2014,

Answer: A

The clock strikes 156 times in 24 hours.

In each 12-hour cycle it strikes 1 + 2 + ... + 12 times, and a day has two such cycles.

\[ \begin{aligned} 1 + 2 + \cdots + 12 &= \frac{12 \times 13}{2} = 78 \\ 2 \times 78 &= 156 \end{aligned} \]

144 is \(12 \times 12\), which treats every hour as 12 strokes, and 164 and 168 match no counting rule. The key step is remembering that the 1-to-12 cycle happens twice a day.

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84.

The sum of all odd numbers from 1 to 51 is

A.676
B.625
C.576
D.696

Exam Year(s)

Tripura Public Service Commission - 2019,

Answer: A

The sum is 676.

The odd numbers from 1 to 51 are 1, 3, 5, ..., 51. There are \(\frac{51 + 1}{2} = 26\) of them, and the sum of the first \(n\) odd numbers is \(n^{2}\):

\[ 1 + 3 + 5 + \cdots + 51 = 26^{2} = 676 \]

625 is \(25^{2}\), the sum up to 49, which leaves out 51. 576 is \(24^{2}\), and 696 does not come from the formula.

The formula works because adding consecutive odd numbers builds perfect squares: 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and so on.

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85.

How many terms are there in the series 201, 208, 215, ..... 369 ?

A.25
B.26
C.23
D.24

Exam Year(s)

Tripura Public Service Commission - 2019,

Answer: A

The series has 25 terms.

201, 208, 215, ... is an arithmetic progression with first term 201 and common difference 7. If 369 is the \(n\)th term:

\[ \begin{aligned} T_n &= a + (n-1)d \\ 369 &= 201 + (n-1) \times 7 \\ n - 1 &= 168 \div 7 = 24 \\ n &= 25 \end{aligned} \]

24 is the number of gaps between terms, the answer you get by forgetting to add 1 for the first term. 26 and 23 are off-by-one slips in the other direction.

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Tags: A.P. G.P. H.P. Progressions Questions and Answers || Arithmetic Progression MCQ Questions || Geometric Progression GK Questions || Harmonic Progression MCQ || Quantitative Aptitude Progressions GK

Progressions (A.P., G.P., H.P.) Questions and Answers

Important Instructions

Topics

Other Subjects

Tags: A.P. G.P. H.P. Progressions Questions and Answers || Arithmetic Progression MCQ Questions || Geometric Progression GK Questions || Harmonic Progression MCQ || Quantitative Aptitude Progressions GK

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