https://file.aiquickdraw.com/mathbot/file/1729000358078ksz08jca.jpg

Let's break down the given problem from the image.

### Given Information:
\[
\sum_{k=1}^{n} (5k - 1) = \frac{n(5n + 3)}{2}
\]

### Part (a):
We are asked to express \(\sum_{k=1}^{n} k\) (the sum of the first \(n\) integers) in terms of \(n\), using the formula provided.

#### Solution:
From the given summation:
\[
\sum_{k=1}^{n} (5k - 1) = 5 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1
\]
We know:
- \(\sum_{k=1}^{n} k\) is the sum of the first \(n\) natural numbers, and
- \(\sum_{k=1}^{n} 1 = n\) because adding 1 for \(n\) terms results in \(n\).

So the equation becomes:
\[
5 \sum_{k=1}^{n} k - n = \frac{n(5n + 3)}{2}
\]
Now, solve for \(\sum_{k=1}^{n} k\):
\[
5 \sum_{k=1}^{n} k = \frac{n(5n + 3)}{2} + n
\]
Combine terms on the right-hand side:
\[
5 \sum_{k=1}^{n} k = \frac{n(5n + 3) + 2n}{2} = \frac{n(5n + 5)}{2} = \frac{5n(n + 1)}{2}
\]
Finally, divide both sides by 5:
\[
\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}
\]
This is the standard formula for the sum of the first \(n\) integers.

### Part (b):
We need to evaluate \(\sum_{k=3}^{100} k\).

This can be broken down as:
\[
\sum_{k=3}^{100} k = \sum_{k=1}^{100} k - \sum_{k=1}^{2} k
\]
Using the formula \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\):
\[
\sum_{k=1}^{100} k = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = 5050
\]
\[
\sum_{k=1}^{2} k = \frac{2(2+1)}{2} = 3
\]
Thus:
\[
\sum_{k=3}^{100} k = 5050 - 3 = 5047
\]

### Final Answers:
(a) \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\)

(b) \(\sum_{k=3}^{100} k = 5047\)

Would you like further details or have any questions?

#### Related Questions:
1. What is the formula for the sum of squares of the first \(n\) integers?
2. How would the solution change if the sum involved \(k^2\) terms instead of \(k\)?
3. How can the formula for \(\sum_{k=1}^{n} k\) be derived geometrically?
4. How does the summation change if a different arithmetic sequence is used?
5. What if the summation was starting from a different index, say \(k=5\)?

#### Tip:
When working with summations, identifying the structure of the sum (arithmetic, geometric, etc.) is crucial for applying the correct formula efficiently.

It is given that ∑(5k - 1) from k=1 to n is equal to (n(5n + 3))/2. (a) Using the formula, express ∑k from k=1 to n in terms of n. (b) Using the result of (a), evaluate ∑k from k=3 to 100.

Learn how to express the summation of ∑k from k=1 to n in terms of n, and evaluate ∑k from k=3 to 100. This problem involves key concepts of arithmetic progression and summations, suitable for students in Grades 9-12.

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