High School

Use Gauss' method to determine the sum of all the whole numbers from 27 to 172.

[tex]27 + 28 + 29 + \ldots + 170 + 171 + 172[/tex]

Answer :

To determine the sum of all the whole numbers from 27 to 172 using Gauss' method, we can follow these steps:

1. Identify the First and Last Terms:
- The first term of the sequence is 27.
- The last term of the sequence is 172.

2. Calculate the Number of Terms:
To find the total number of terms in this sequence, subtract the first term from the last term, and then add 1:
[tex]\[
\text{Number of terms} = 172 - 27 + 1 = 146
\][/tex]

3. Apply Gauss' Formula:
Gauss' method tells us that the sum of an arithmetic sequence can be calculated using the formula:
[tex]\[
\text{Sum} = \frac{\text{Number of terms} \times (\text{First term} + \text{Last term})}{2}
\][/tex]
Plug in the values we have:
[tex]\[
\text{Sum} = \frac{146 \times (27 + 172)}{2}
\][/tex]

4. Calculate the Sum:
First, calculate the expression inside the parentheses:
[tex]\[
27 + 172 = 199
\][/tex]
Then multiply by the number of terms:
[tex]\[
146 \times 199 = 29054
\][/tex]
Finally, divide by 2 to get the total sum:
[tex]\[
\text{Sum} = \frac{29054}{2} = 14527
\][/tex]

So, the sum of all the whole numbers from 27 to 172 is 14,527.

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