UPSC Prelims 2008 Q2: Science & Technology – Quantitative Aptitude
What is the number of terms in the series 117, 120, 123, 126, ..., 333?
- (a) 72
- (b) 73
- (c) 76
- (d) 79
Answer: (b) 73
Explanation
Counting terms in an arithmetic series
The series 117, 120, 123, ..., 333 increases by a constant common difference.
| Step | Calculation |
|---|---|
| Common difference | 120 − 117 = 3 |
| Formula | Number of terms = (Last − First)/Difference + 1 |
| Substitute | (333 − 117)/3 + 1 = 216/3 + 1 = 72 + 1 = 73 |
Why we add 1:
| Point | Detail |
|---|---|
| Counting both ends | The formula counts the number of steps (72) plus the starting term itself |
| Common error | Stopping at 72 forgets that the first term (117) is itself one of the terms being counted |
Answer: (b) 73.
Why each option is right or wrong
- (a) Incorrect. 72 is the number of steps/intervals, but forgets to add 1 for the starting term.
- (b) CORRECT. (333−117)/3 + 1 = 73 terms total.
- (c) Incorrect. 76 overstates the correct count.
- (d) Incorrect. 79 also overstates the correct count.
Key point
For an arithmetic series, number of terms = (Last term − First term)/Common difference + 1. Here: (333−117)/3 + 1 = 73.
Remember
Series 117 to 333, common difference 3: total terms = 73.
Common mistakes
- Forgetting to add 1 after dividing the range by the common difference.
- Miscalculating the common difference between consecutive terms.
- Making a simple division error in the arithmetic.
Exam tip
Arithmetic series term count = (Last − First)/Difference + 1. Always remember the '+1' for the starting term.
Sources
- NCERT Class X Mathematics — Arithmetic Progressions
- NCERT Class XI Mathematics — Sequences and Series