Q.42 The average of all positive even integers less than or equal to 40 is ___.
Step-by-Step Solution
The average of all positive even integers less than or equal to 40 is 21. Positive even integers ≤ 40 form the arithmetic sequence: 2, 4, 6, …, 40.
First term \( a = 2 \), common difference \( d = 2 \), last term \( l = 40 \). Number of terms \( n = \frac{l – a}{d} + 1 = 20 \).
Sum \( S = \frac{n}{2} (a + l) = \frac{20}{2} \times 42 = 420 \). Average = \( \frac{S}{n} = \frac{420}{20} = 21 \).
Common Options Explained
- 20: Matches midpoint of 2 to 38 (excludes 40); incorrect as sequence reaches 40.
- 21: Correct; equals \( \frac{2 + 40}{2} \) for arithmetic sequence average.
- 22: Assumes 4 to 40 (19 terms); wrong count.
- 23/24: Overestimates by miscounting terms or using odd sequence formula.
Quick Calculation Method
The average of all positive even integers less than or equal to 40 is a key arithmetic progression question often appearing in competitive exams like CSIR NET. This calculation tests understanding of sequences and averages.
Even numbers ≤ 40: 2 to 40 (20 terms). Average = \( \frac{\text{first} + \text{last}}{2} = 21 \).
Verify sum: \( 20 \times 21 = 420 \). Dividing confirms precision.
Why 21 Beats Other Options
| Option | Calculation Error | Corrected Value |
|---|---|---|
| 20 | Ends at 38 | 21 |
| 21 | Full sequence | 21 |
| 22 | Wrong term count | 21 |
| 23/24 | Inflated midpoint | 21 |
Master this for quantitative aptitude sections. Practice similar: average up to 100 is 51. [memory:3]
