Question 1144108: find the sum 48+49+50+51+...+146+147
Answer by greenestamps(13200)   (Show Source):
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The average of any set of numbers is the sum of the numbers, divided by how many there are.
Therefore, the sum of any set of numbers is the number of numbers in the set, multiplied by the average of the numbers.
In an arithmetic sequence like this (constant difference between one number and the next), it is easy to find both the number of numbers and the average, so it is easy to find the sum.
With a common difference of 1, the number of numbers is simply (last minus first) plus 1: (147-48)+1 = 99+1 = 100.
In any arithmetic sequence, the average of all the numbers is the average of the first and last numbers. So in this sum the average is (48+147)/2 = 195/2 = 97.5.
And then the sum of the numbers is the sum of 100 numbers with an average of 97.5: 100(97.5) = 9750.
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