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.