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Average of Consecutive Numbers

Introduction

A very common type of average problem in aptitude exams involves consecutive numbers - natural numbers, even numbers, or odd numbers. These problems are simple once you know the property that the average of consecutive numbers is equal to their middle term.

This property helps you solve such problems quickly without adding all numbers.

Pattern: Average of Consecutive Numbers

Pattern

The average of consecutive numbers (whether natural, even, or odd) is always the middle number.

- If there are an odd count of numbers → Average = Middle number.
- If there are an even count of numbers → Average = Mean of the two middle numbers.

For a sequence of consecutive natural numbers from 1 to n: Average = (n + 1) ÷ 2.

For consecutive even or odd numbers: Average = (First + Last) ÷ 2.

Step-by-Step Example

Question

Find the average of the first 10 natural numbers.

Options:

  • A. 5.5
  • B. 6
  • C. 4.5
  • D. 5

Solution

  1. Step 1: Understand the sequence

    First 10 natural numbers = 1, 2, 3, …, 10.

  2. Step 2: Apply the formula

    Average of first n natural numbers = (n + 1) ÷ 2.

  3. Step 3: Substitute the value of n

    Average = (10 + 1) ÷ 2 = 11 ÷ 2 = 5.5.

  4. Final Answer:

    5.5 → Option A

  5. Quick Check:

    Middle numbers are 5 and 6 → (5 + 6) ÷ 2 = 5.5 ✅

Quick Variations

- Average of first 20 natural numbers = (20 + 1)/2 = 10.5.

- Average of consecutive even numbers from 2 to 20 = (2 + 20)/2 = 11.

- Average of consecutive odd numbers from 1 to 19 = (1 + 19)/2 = 10.

Trick to Always Use

  • Average of consecutive numbers = middle term.
  • If count is even → take mean of two middle terms.
  • For first n natural numbers → formula (n + 1)/2.
  • Shortcut saves time compared to adding all numbers.

Summary

Summary

The Average of Consecutive Numbers is solved using the middle number property or formula.

  • Odd count: Average = Middle number.
  • Even count: Average = Mean of two middle numbers.
  • 1 to n: Average = (n + 1)/2.
  • Consecutive even/odd numbers: Average = (First + Last)/2.

Practice

(1/5)
1. Find the average of the first 15 natural numbers.
easy
A. 8
B. 7.5
C. 8.5
D. 7

Solution

  1. Step 1: Recall the formula

    Formula for average of first n natural numbers = (n + 1) ÷ 2.

  2. Step 2: Substitute n

    For n = 15 → (15 + 1) ÷ 2 = 16 ÷ 2 = 8.

  3. Final Answer:

    8 → Option A

  4. Quick Check:

    The middle term of 1..15 is 8 → confirms result ✅
Hint: Use (n + 1) ÷ 2 for 1 to n.
Common Mistakes: Adding all numbers manually instead of using the formula.
2. What is the average of consecutive even numbers from 2 to 20?
easy
A. 11
B. 10
C. 12
D. 13

Solution

  1. Step 1: Use sequence property

    For an arithmetic sequence, average = (first + last) ÷ 2.

  2. Step 2: Compute

    (2 + 20) ÷ 2 = 22 ÷ 2 = 11.

  3. Final Answer:

    11 → Option A

  4. Quick Check:

    Middle terms are 10 and 12 → (10 + 12) ÷ 2 = 11 ✅
Hint: Use (first + last)/2 for any consecutive sequence.
Common Mistakes: Trying to average all terms individually instead of using first+last.
3. What is the average of consecutive even numbers between 40 and 60?
easy
A. 48
B. 49
C. 51
D. 50

Solution

  1. Step 1: Identify endpoints

    First even = 40, last even = 60.

  2. Step 2: Compute average

    Average = (40 + 60) ÷ 2 = 100 ÷ 2 = 50.

  3. Final Answer:

    50 → Option D

  4. Quick Check:

    Middle even number is 50 → confirms result ✅
Hint: Average = (first + last)/2 for arithmetic sequences.
Common Mistakes: Counting terms incorrectly or picking wrong endpoints.
4. Find the average of the first 25 odd numbers.
medium
A. 24
B. 25
C. 26
D. 27

Solution

  1. Step 1: Identify first and last

    The nth odd number = 2n - 1; for n = 25 the last odd is 49.

  2. Step 2: Apply average formula

    Average = (first + last) ÷ 2 = (1 + 49) ÷ 2 = 50 ÷ 2 = 25.

  3. Final Answer:

    25 → Option B

  4. Quick Check:

    The 13th odd number is 25 → matches average ✅
Hint: For first n odd numbers the average equals n.
Common Mistakes: Using arithmetic mean of first/last incorrectly or off-by-one errors.
5. The average of consecutive numbers from 50 to 100 is?
medium
A. 74
B. 76
C. 75
D. 73

Solution

  1. Step 1: Apply range average

    Average for a range a to b = (a + b) ÷ 2.

  2. Step 2: Compute

    (50 + 100) ÷ 2 = 150 ÷ 2 = 75.

  3. Final Answer:

    75 → Option C

  4. Quick Check:

    75 is the middle number between 50 and 100 → confirms result ✅
Hint: Range average = (first + last)/2.
Common Mistakes: Dividing by number of terms instead of using the first+last shortcut.

Mock Test

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