Formula for Adding Consecutive Numbers
Before going to learn what is the formula for adding consecutive numbers, first, we will recall what are consecutive numbers. The consecutive numbers are numbers that follow each other in increasing order. Two consecutive numbers always differ by 1.
Examples of Consecutive Numbers are:12, 13, 14, ..., or 2, 1, 0, 1, 2, .... Since the difference between every two consecutive numbers is the same (as 1), every list of consecutive numbers is an arithmetic sequence. So the sum of the arithmetic sequence formula can be used as the formula for adding consecutive numbers.
What is the Formula for Adding Consecutive Numbers?
We know that the sum of an arithmetic sequence. of n terms, a + (a + d) + (a + 2d) + ... + (a + (n1) d) is: Sum of n terms = (n/2) (first term + last term)
We already have seen in the earlier section that, the difference between any two consecutive numbers is, d = 1. So the formula for adding n consecutive integers [ a + (a + 1) + (a + 2) + .... {a + (n1)}] is,
Sum of n consecutive numbers = (n/2) (first number + last number)
Solved Examples Using Formula for Adding Consecutive Numbers

Example 1: Find the sum: 46 + 47 + 48 + ... + 103 by using the formula for adding consecutive numbers.
Solution:
To find: The sum 46 + 47 + 48 + ... + 103.
Here, the number of numbers is, n = 103  46 + 1 = 58.
First number = 46.
Last number = 103.
Using the sum of consecutive numbers formula:
Sum of n consecutive numbers = \(\frac{n}{2}\) (first number + last number)
Sum of 58 consecutive numbers = \(\frac{58}{2}\) (46 + 103) = 4321
Answer: 46 + 47 + 48 + ... + 103 = 4,321.

Example 2: Find the sum of all integers from 10 to 210 by using the formula for adding consecutive numbers.
Solution:
To find: The sum of integers from 10 to 210.
Here, the first number = 10.
Last number = 210.
Number of integers from 10 to 210 is, n = 210  (10) + 1 = 221
Using the sum of consecutive numbers formula:
Sum of n consecutive numbers = \(\frac{n}{2}\) (first number + last number)
Sum of 58 consecutive numbers = \(\frac{221}{2}\) (10 + 210) = 22100
Answer: The sum of integers from 10 to 210 is 22,100.