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What is the average of all the natural numbers from 1 to 98?
49
50
50.5
49.5
49
For any arithmetic progression where the difference between terms is constant, the average is simply the mean of the first and last term.
Range of natural numbers from 1 to 98 (inclusive). First term a = 1, last term l = 98.
Average=2a+lтАЛ
For any arithmetic progression where the difference between terms is constant, the average is simply the mean of the first and last term.
Many students mistakenly count the number of terms as 98 - 1 = 97, forgetting that the range is inclusive, making the number of terms n = 98.
Identify the Sequence
The numbers 1, 2, 3,..., 98 form an arithmetic progression where the first term a=1 and the last term l=98.
a=1,l=98
Apply Average Formula
For any arithmetic sequence, the average is calculated as the sum of the first and last terms divided by two.
Average=21+98тАЛ
Perform Final Calculation
Calculate the sum and divide by 2 to get the result: 99/2=49.5.
Average=49.5
D is correct because the average of an arithmetic series is the mean of the first and last terms, calculated as 99/2 = 49.5.
This concept is foundational for calculating the Mean in Statistics and understanding sum-of-series formulas in Algebra.
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