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What is the average of all the natural numbers from 1 to 97?
48.5
49
50
49.5
49
For any arithmetic progression with a constant difference, the average is simply the middle value, calculated as (First + Last) / 2.
Natural numbers from 1 to 97. The series is 1, 2, 3, ..., 97.
Average=2First┬аTerm+Last┬аTermтАЛ
For any arithmetic progression with a constant difference, the average is simply the middle value, calculated as (First + Last) / 2.
Students often mistake the average for the sum of numbers or incorrectly count the number of terms as 96 instead of 97.
Identify the Series
The numbers 1, 2, 3,..., 97 form an Arithmetic Progression (AP) where the first term a=1 and the last term l=97.
a=1,l=97
Apply Average Formula
For any AP, the average is the mean of the first and last terms.
Avg=2a+lтАЛ
Calculate Result
Substituting the values into the formula gives 21+97тАЛ=298тАЛ=49.
Avg=49
B is correct because the average of a consecutive sequence of natural numbers is the arithmetic mean of the first and last values, resulting in 49.
This formula applies to any sequence in AP (e.g., odd numbers, multiples of 5), making it a high-utility shortcut for quick calculations in Data Interpretation and Speed Math sections.
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