2.2 Average of Consecutive Numbers

(a) n consecutive numbers (any consecutive integers): Average = (First term + Last term)/(2)

This works because consecutive numbers form an Arithmetic Progression (AP), and the average of any AP equals the average of its first and last terms (or equivalently, the middle term when the number of terms is odd).

(b) Average of first n natural numbers: Average = (n+1)/(2)

(c) Average of first n consecutive even numbers: Average = n+1

(d) Average of first n consecutive odd numbers: Average = n

Solved Example 4: Find the average of the first 50 natural numbers.

Average = (50 + 1)/2 = 51/2 = 25.5.

Solved Example 5: Find the average of the first 30 even natural numbers.

Average = n + 1 = 30 + 1 = 31. (Check: first 30 even numbers are 2, 4, …, 60. Average = (2+60)/2 = 31. ✓)

Solved Example 6: Find the average of the first 25 odd natural numbers.

Average = n = 25. (Check: numbers are 1, 3, …, 49. Average = (1+49)/2 = 25. ✓)

Exam tip: these three formulas (n+1)/2, n+1, and n are frequently confused with each other under time pressure. Memorise them as a set: “natural → half of (n+1); even → whole of (n+1); odd → just n.”

Extension — average of consecutive numbers NOT starting from 1: For any AP (including consecutive integers, evens, or odds that do not start at 1), always fall back on the universal rule: Average = (First term + Last term)/2. This single formula subsumes all the special cases above and should be your default tool whenever the starting point is not 1.

Solved Example 6(a): Find the average of all even numbers from 20 to 60 (inclusive).

Average = (First + Last)/2 = (20+60)/2 = 80/2 = 40.

Solved Example 6(b): Find the average of all odd numbers between 30 and 70.

Odd numbers strictly between 30 and 70: 31, 33, …, 69. Average = (31+69)/2 = 100/2 = 50.

Solved Example 6(c): Find the average of numbers from 15 to 45 (inclusive).

Average = (15+45)/2 = 60/2 = 30. (Count of numbers, if needed separately, = 45−15+1 = 31, but note the average formula does not require the count at all — this is what makes it so fast.)

Solved Example 6(d): Find the average of all multiples of 7 between 1 and 100.

The multiples of 7 in this range run from 7 up to 98 (since 7×14=98 and 7×15=105 exceeds 100). Average = (First+Last)/2 = (7+98)/2 = 105/2 = 52.5.

Solved Example 6(e): Find the average of 12 consecutive multiples of 4, the smallest of which is 8.

The 12 terms are 8, 12, 16, …, up to the 12th term = 8 + (12−1)×4 = 8+44 = 52. Average = (First+Last)/2 = (8+52)/2 = 60/2 = 30.