CDS Mathematics · Averages

Averages of Consecutive Numbers

Equally spaced numbers average to their middle term, which is also the average of the first and the last.

All Averages notes CDS Mathematics strategy

Why this matters

Eight PYQs, one of them HARD. Consecutive integers, consecutive evens, 11 to 99, the first hundred odd numbers: each is equally spaced, so the average is the middle term and no adding is needed. The one list that is NOT equally spaced, the composite numbers, has to be written out.

Concept 1 of 1: The middle term

In an equally spaced list, the terms pair off from the two ends, each pair averaging to the middle. So the whole list averages to the middle term, or to half of first plus last.

Definition

Equally spaced: average $= \dfrac{\text{first} + \text{last}}{2} =$ the middle term.
$n$ consecutive integers averaging $A$ : largest $= A + \dfrac{n - 1}{2}$ .
Extending $n$ consecutive numbers by $k$ more raises the average by $\dfrac k2$ .
First $n$ odd numbers: average $n$ . First $n$ even numbers: average $n + 1$ .
Lists that are not equally spaced (primes, composites): write them out and add.

Equally spaced list

\text{average} = \dfrac{\text{first} + \text{last}}{2}

Worked example

The average of

7

consecutive integers is

40

. Find the largest.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q44Easy

Example 1 · Averages · Averages of Consecutive Numbers

If the average of 9 consecutive positive integers is 55, then what is the largest integer?

Only for equally spaced lists

The first ten composite numbers

4, 6, 8, 9, 10, 12, 14, 15, 16, 18

are not equally spaced; first plus last over two gives

11

, but the true average is

11.2

Drill 7 more on the middle term

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The middle term
Equally spaced list
$\text{average} = \dfrac{\text{first} + \text{last}}{2}$

Watch out for (1)

Only for equally spaced lists
→ The middle term

Test yourself on Averages

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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