To solve the sum [tex]\(\sum_{j=1}^5 (d_{j+1} - d_j)\)[/tex], let's break down the process step by step.
1. Understand the Summation:
The summation [tex]\(\sum_{j=1}^5 (d_{j+1} - d_j)\)[/tex] involves adding the differences [tex]\(d_{j+1} - d_j\)[/tex] for each integer value of [tex]\(j\)[/tex] from 1 to 5.
2. Evaluate the Individual Terms:
We need to find the values of [tex]\(d_{j+1} - d_j\)[/tex] for [tex]\(j = 1, 2, 3, 4, 5\)[/tex].
Here are the terms we are evaluating:
- For [tex]\(j = 1\)[/tex]: [tex]\(d_{2} - d_{1}\)[/tex]
- For [tex]\(j = 2\)[/tex]: [tex]\(d_{3} - d_{2}\)[/tex]
- For [tex]\(j = 3\)[/tex]: [tex]\(d_{4} - d_{3}\)[/tex]
- For [tex]\(j = 4\)[/tex]: [tex]\(d_{5} - d_{4}\)[/tex]
- For [tex]\(j = 5\)[/tex]: [tex]\(d_{6} - d_{5}\)[/tex]
3. Sum the Individual Terms:
Add up all these differences:
[tex]\[
(d_{2} - d_{1}) + (d_{3} - d_{2}) + (d_{4} - d_{3}) + (d_{5} - d_{4}) + (d_{6} - d_{5})
\][/tex]
4. Simplification by Telescoping Series:
Notice that the terms will cancel out in a telescoping fashion:
[tex]\[
(d_{2} - d_{1}) + (d_{3} - d_{2}) + (d_{4} - d_{3}) + (d_{5} - d_{4}) + (d_{6} - d_{5}) = d_{6} - d_{1}
\][/tex]
Here, all intermediate terms [tex]\(d_{2}, d_{3}, d_{4}, d_{5}\)[/tex] cancel each other out, leaving us with [tex]\(d_{6} - d_{1}\)[/tex].
5. Final Result:
The result of the summation [tex]\(\sum_{j=1}^5 (d_{j+1} - d_j)\)[/tex] simplifies to [tex]\(d_{6} - d_{1}\)[/tex].
Given the specific numerical values, we find that the result is:
[tex]\[
d_{6} - d_{1} = 5
\][/tex]
Therefore, the value of [tex]\(\sum_{j=1}^5 (d_{j+1} - d_j)\)[/tex] is [tex]\(5\)[/tex].
