Noticing Patterns in Addition


Question Description

What is the sum of integers from 1 to n?Justify your formula with a proof.

Tutor Answer

School: UC Berkeley

Before answering the question, you have to understand what integers are. Integers are numbers which are not fractions or decimals. In simple words, these are whole numbers. These numbers may be positive or negative. But since the question only pertains to positive 1 to any number, n, we will only deal with positive integers. Since we’re dealing with the sum of positive whole numbers, expect a result which is also a positive whole number. The formula in getting the sum of integers is: S = (n/2)(n+1) where, S = sum of integers n = any positive number = the highest number to be added = the last number in the series Proof: Using the formula given, look for the sum of integers from 1 to 10. S = (10/2) (10+1) S = 5 (11) S = 55 Adding the numbers 1,2,3,4,5,6,7,8,9,10, manually, would result to 55. This method proves that our formula is correct. Looking at the formula in a different perspective, divide the given numbers into 2 equal groups: the lower half and the higher half. This would result to: Lower Half Higher Half 1 2 3 4 5 ...

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