How to Find the Average of Consecutive Numbers?
Average of Consecutive Numbers is found by using the middle term for an odd number of equally spaced terms, or the midpoint of the two middle terms for an even number of terms. This page covers the average of consecutive integers, consecutive even and odd numbers, standard formulas, and short methods for solving aptitude questions.
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What Is the Average of Consecutive Numbers?
If the first term is a and the last term is l, the average is (a + l) ÷ 2. For an odd number of consecutive terms, the average is the middle term. For an even number of terms, it lies halfway between the two middle terms. For example, the average of 12, 13, 14, 15 and 16 is 14.
Average of Consecutive Numbers 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 Consecutive Numbers Formula & Tricks
Important Formulas
Add all the terms and divide the sum by their count.
This applies to any equally spaced sequence, including consecutive integers, even numbers and odd numbers.
Here, a is the first integer and n is the number of terms. The last term is a + n − 1.
Here, a is the first term and d is the common difference. For consecutive even or odd numbers, d = 2.
If the average and number of terms are known, their product gives the total sum.
Quick Tricks
For an odd number of equally spaced terms, select the middle term directly instead of adding every number.
In an equally spaced sequence, the sum of the first and last terms equals the sum of the second and second-last terms. Therefore, the average is the midpoint of the first and last terms.
If equal numbers of terms are removed from both ends, the average remains unchanged. If terms are removed only from one side, recalculate using the new first and last terms.
Average of Consecutive Numbers Concepts
Average of Consecutive Integers
If the terms are a, a + 1, a + 2, ..., a + n − 1, then average = [a + (a + n − 1)] ÷ 2 = a + (n − 1) ÷ 2. When n is odd, this value is an integer and is the middle term. When n is even, it may be a fraction or end in .5.
Average of Consecutive Even Numbers
For n terms beginning with a, the sequence is a, a + 2, a + 4, ..., a + 2(n − 1). Its average is [a + a + 2(n − 1)] ÷ 2 = a + n − 1.
Average of Consecutive Odd Numbers
For n terms beginning with a, the sequence is a, a + 2, a + 4, ..., a + 2(n − 1). The average is a + n − 1. With an odd number of terms, the average is the central odd number.
Finding the Number of Terms
For equally spaced terms with common difference d, the number of terms is n = [(Last term − First term) ÷ d] + 1. For consecutive even or odd numbers, use d = 2.
Finding Consecutive Numbers from Their Average
For an odd number of consecutive integers, the average is the middle integer. For n terms with common difference d, the first term is Average − [(n − 1)d ÷ 2], and the last term is Average + [(n − 1)d ÷ 2].
Average of Consecutive Numbers Data Table
Use the structured data below while following the explanation and examples.
Examples by Number Type
The first and last terms determine the average of each equally spaced sequence.
| Number type | Terms | Calculation | Average |
|---|---|---|---|
| Consecutive integers | 7, 8, 9, 10, 11 | (7 + 11) ÷ 2 | 9 |
| Consecutive even numbers | 12, 14, 16, 18 | (12 + 18) ÷ 2 | 15 |
| Consecutive odd numbers | 17, 19, 21, 23, 25 | (17 + 25) ÷ 2 | 21 |
Note: The common difference is 1 for consecutive integers and 2 for consecutive even or odd numbers.
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Average of Consecutive Numbers: Revision Points
Use these formulas and rules for quick revision.
- Average of equally spaced terms = (First term + Last term) ÷ 2.
- For n consecutive integers beginning with a, average = a + (n − 1) ÷ 2.
- For n consecutive even or odd numbers beginning with a, average = a + n − 1.
- Number of consecutive integers = Last term − First term + 1.
- Number of equally spaced terms = [(Last term − First term) ÷ Common difference] + 1.
- For an odd number of terms, the average is the middle term.
- For an even number of terms, the average is halfway between the two middle terms.
- Sum of terms = Number of terms × Average.
Average of Consecutive Numbers FAQs
What is the average of the first 10 natural numbers?
The numbers are 1 to 10. Average = (1 + 10) ÷ 2 = 5.5.
What is the average of 20 consecutive integers starting from 11?
The last integer is 30. Average = (11 + 30) ÷ 2 = 20.5.
What is the average of the first eight even numbers?
The first eight positive even numbers are 2 to 16. Average = (2 + 16) ÷ 2 = 9.
Can the average of consecutive integers be a decimal?
Yes. An even number of consecutive integers has an average ending in .5, such as the average of 4, 5, 6 and 7, which is 5.5.
What is the average of 7 consecutive odd numbers with middle term 25?
The average is the middle term, so it is 25. The numbers are 19, 21, 23, 25, 27, 29 and 31.
How many consecutive odd numbers are there from nine to 31?
Using n = [(31 − 9) ÷ 2] + 1, n = 11 + 1 = 12. However, 9 and 31 are both odd and the sequence contains 12 terms: 9, 11, ..., 31.