Calculating the Sum of Natural Numbers from 11 to 90: A Comprehensive Guide

When dealing with sequences and series in mathematics, one often encounters the need to determine the sum of a specific range of natural numbers. This article will provide a detailed explanation of how to calculate the sum of all natural numbers from 11 to 90 using various methods, including the arithmetic series formula and other techniques.

Method 1: Using Arithmetical Series Formula

To find the sum of natural numbers from 11 to 90, we can use the formula for the sum of an arithmetic series:

Sn (n/2) times; (a l)

Where:

n is the number of terms in the series, a is the first term, l is the last term.

Identifying Parameters

In our case, the first term (a) is 11, and the last term (l) is 90. We need to calculate the total number of terms (n).

The number of terms (n) is calculated as:

n l - a 1

Substituting the values of a and l:

n 90 - 11 1 80

Calculating the Sum

Using the formula for the sum of an arithmetic series:

S80 (80/2) times; (11 90) 40 times; 101 4040

Therefore, the sum of all natural numbers from 11 to 90 is 4040.

Method 2: Using the Combined Sum of Two Arithmetical Series

Another approach is to break down the range into two parts and sum them separately. We can find the sum of all natural numbers from 1 to 90 and subtract the sum of all natural numbers from 1 to 10:

Sum from 1 to 90 (90/2) times; (1 90) 40 times; 91 3640

Sum from 1 to 10 (10/2) times; (1 10) 5 times; 11 55

Taking the difference:

4040 3640 - 55

Method 3: Using Arithmetic Progressions and Series Notation

We can represent the series in a more conceptual form:

10111213………8990

Breaking it down into two parts:

0 11213………8990

Sum 1 (11 12 13 … 89 90 - 0)

Sum 1 4040 - 0 4041

However, this method is not correct, and the sum should be 4040, as calculated previously.

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