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What is the average of all the natural numbers from 1 to 91?
46.5
46
45.5
47
46
For any continuous arithmetic progression starting from 1 up to n, the average of natural numbers from 1 to n is simply given by 2n+1тАЛ.
Natural numbers from 1 to 91. First term a=1, last term l=91, and total number of terms n=91.
Average=2n+1тАЛ
For any continuous arithmetic progression starting from 1 up to n, the average of natural numbers from 1 to n is simply given by 2n+1тАЛ.
Students often use the sum formula 2n(n+1)тАЛ to find the sum and then forget to divide by n, or incorrectly divide by nтИТ1.
Identify the given sequence
The series of natural numbers from 1 to 91 forms an arithmetic progression (AP) with a common difference d=1.
n=91,a=1,l=91
Recall the average formula for AP
For any arithmetic progression, the average of the terms is given by the mean of the first and the last term: 2First┬аTerm+Last┬аTermтАЛ.
Average=2a+lтАЛ
Substitute the values
Substituting a=1 and l=91 into the formula: Average=21+91тАЛ.
Average=292тАЛ
Calculate the final value
Dividing 92 by 2 gives the final average value.
Average=46
B is correct because the average of natural numbers from 1 to 91 evaluates to 21+91тАЛ=292тАЛ=46.
The sum of the first n natural numbers is 2n(n+1)тАЛ, and dividing it by n directly yields the average formula 2n+1тАЛ, which is useful across quantitative aptitude topics.