Average of Arithmetic Progression with Solved Examples
Average of Arithmetic Progression is found by taking the mean of the first and last terms. For an AP with first term a and last term l, the average is (a + l)/2. If the number of terms is odd, the average is the middle term; if it is even, it is the mean of the two middle terms.
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What Is the Average of an Arithmetic Progression?
If the first term is a, the last term is l, and there are n terms, then Average = (a + l)/2. Since the terms are equally spaced, terms equidistant from the ends have the same pair sum, a + l. For example, the average of 5, 9, 13, 17, 21 is (5 + 21)/2 = 13.
Average of Arithmetic Progression Video Lessons
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Average of AP and Consecutive Numbers
Learn how to calculate the average of terms in an arithmetic progression and consecutive numbers using their central-term and series-based relationships.
Average of Arithmetic Progression Formula & Tricks
Important Formulas
Here, a is the first term and l is the last term of the arithmetic progression.
The last term is l = a + (n - 1)d, where n is the number of terms and d is the common difference.
Sₙ is the sum of n terms. Since Sₙ = n[2a + (n - 1)d] / 2, this gives the AP average formula.
When an AP has an odd number of terms, the middle term is at position (n + 1)/2 and equals the average.
Quick Tricks
The common difference and number of terms are not required when both end terms are known. Add the first and last terms and divide by 2.
In an AP, the first and last terms have the same sum as the second and second-last terms, and so on. Therefore, the overall average is the mean of the two end terms.
For an odd number of equally spaced terms, the central term is the average. Its position is (n + 1)/2.
Average of Arithmetic Progression Concepts
Average from the First and Last Terms
This rule works for increasing and decreasing APs. It follows from the equal spacing of consecutive terms: each term can be paired with a term equally distant from the opposite end, and every pair has sum a + l.
Average When the Number of Terms Is Odd
The position of the middle term is (n + 1)/2. The terms on either side are equally distant from this term, so their deviations cancel.
Average When the Number of Terms Is Even
The two central positions are n/2 and (n/2) + 1. Their mean is equal to (a + l)/2 because they are equally spaced around the centre of the progression.
Average When Terms Are Given by a and d
The nth term is a + (n - 1)d. Substituting this as the last term in (a + l)/2 gives the direct formula. The value of d may be positive, zero or negative.
Average of Arithmetic Progression Data Table
Use the structured data below while following the explanation and examples.
Average Rules for an Arithmetic Progression
The correct average rule depends on the number and form of the given AP terms.
| Given situation | Rule | Example | Average |
|---|---|---|---|
| First and last terms known | (a + l)/2 | 12, ..., 40 | 26 |
| Odd number of terms | Middle term | 5, 9, 13, 17, 21 | 13 |
| Even number of terms | Mean of two middle terms | 4, 8, 12, 16, 20, 24 | 14 |
| a, d and n known | [2a + (n - 1)d]/2 | a = 6, d = 4, n = 5 | 14 |
Note: In the last row, the AP is 6, 10, 14, 18, 22, so its average is (6 + 22)/2 = 14.
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Average of Arithmetic Progression: Revision Points
Use these rules to calculate the average of a finite arithmetic progression.
- Average = (first term + last term)/2.
- Average = [2a + (n - 1)d]/2 when a, d and n are given.
- For an odd number of terms, the average is the middle term.
- For an even number of terms, the average is the mean of the two middle terms.
- The formula works for increasing, constant and decreasing arithmetic progressions.
- The average of n AP terms is Sₙ/n, where Sₙ is their sum.
Average of Arithmetic Progression FAQs
What is the average of an arithmetic progression formula?
For first term a and last term l, Average = (a + l)/2. If a, common difference d and number of terms n are given, Average = [2a + (n - 1)d]/2.
What is the average of the AP 3, 7, 11, 15, 19?
The first term is 3 and the last term is 19. Average = (3 + 19)/2 = 11, which is also the middle term.
How do you find the average of an AP with an even number of terms?
Take the mean of the two middle terms. For 10, 14, 18, 22, the middle terms are 14 and 18, so the average is (14 + 18)/2 = 16.
Does the average formula work for a decreasing arithmetic progression?
Yes. For 50, 45, 40, 35, 30, the average is (50 + 30)/2 = 40.
What is the average of an AP when the first term is 9, common difference is 5 and there are 8 terms?
Average = [2(9) + (8 - 1)(5)]/2 = 53/2 = 26.5.
Is the average of an AP always one of its terms?
It is a term when the AP has an odd number of terms. With an even number of terms, the average is the mean of the two middle terms and may not itself be a term.