Question

How can you find the sum of the first 5 consecutive positive integers? You can easily add the numbers mentally—the sum is 15. But what if you want to find the sum of the first ​n ​consecutive whole numbers? Keep this question in mind and make some conjectures about a formula.

Answer 1

Given:

First five consecutive positive integers.

To find:

The sum of the first 5 consecutive positive integers.

Sum of the first ​n ​consecutive whole numbers.

Solution:

First five consecutive positive integers are 1, 2, 3, 4 and 5.

Sum of these numbers is

$1+2+3+4+5=15$

Therefore, the sum of the first 5 consecutive positive integers is 15.

First ​n ​consecutive whole numbers are 0, 1, 2, 3,..., (n-1). These numbers are in AP.

Here, first term is 0 and common difference is 1.

Sum of n terms of an AP is

$S_n=\dfrac{n}{2}[2a+(n-1)d]$

where, a is first term and d is common difference.

Substitute a=0 and d=1 in the above formula.

$S_n=\dfrac{n}{2}[2(0)+(n-1)(1)]$

$S_n=\dfrac{n}{2}[n-1]$

Sum of first 5 consecutive positive integers is equal to the sum of first 6 consecutive whole number because in whole numbers 0 is extra number.

For n=6,

$S_6=\dfrac{6}{2}[6-1]$

$S_6=3(5)$

$S_6=15$

The sum of the first 5 consecutive positive integers is 15.

So, the sum of (n-1) consecutive positive integers is

$S_n=\dfrac{n}{2}[n-1]$