To solve the summation [tex]\(\sum_{k=7}^{26}\left(\frac{1}{10}\right)\)[/tex], we should first recognize the structure of the summation. Here's a step-by-step guide to the solution:
1. Understanding the Summation:
- The summation [tex]\(\sum_{k=7}^{26}\left(\frac{1}{10}\right)\)[/tex] means we are adding [tex]\(\frac{1}{10}\)[/tex] for each integer [tex]\(k\)[/tex] from 7 to 26, inclusive.
2. Identifying the Terms:
- Notice that each term of the summation is a constant value [tex]\(\frac{1}{10}\)[/tex].
- Hence, the summation simplifies to adding [tex]\(\frac{1}{10}\)[/tex] repeatedly for each integer value of [tex]\(k\)[/tex] from 7 to 26.
3. Counting the Number of Terms:
- The number of terms in the summation is determined by the range of values for [tex]\(k\)[/tex]. To find the number of terms:
- Subtract the lower limit (7) from the upper limit (26) and add 1 (since we include both endpoints).
- The calculation is: [tex]\(26 - 7 + 1 = 20\)[/tex].
- Therefore, there are 20 terms in the summation.
4. Computing the Summation:
- Now that we know there are 20 terms, each being [tex]\(\frac{1}{10}\)[/tex]:
- Multiply the number of terms by the value of each term:
[tex]\[
20 \times \frac{1}{10} = 2
\][/tex]
So, the value of the summation [tex]\(\sum_{k=7}^{26}\left(\frac{1}{10}\right)\)[/tex] is [tex]\(\boxed{2}\)[/tex].
Let's recap:
- We identified the summation as adding [tex]\(\frac{1}{10}\)[/tex] for each [tex]\(k\)[/tex] from 7 to 26.
- We counted that there are 20 terms in total.
- Multiplying 20 by [tex]\(\frac{1}{10}\)[/tex] gives us the final result of 2. Thus, the detailed solution confirms that the sum is [tex]\(2\)[/tex].
