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An arithmetic series is a sum of numbers with a constant difference between each consecutive term. It is one of the simplest types of series used in mathematics. For example, in part (a) of the exercise, where the summation runs from 9 to 36, you can identify it as an arithmetic series because each number increases by the same amount as the previous one.
To find the sum of an arithmetic series, we use the formula:

• \( S = \frac{n}{2} (a + l) \)

Here, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. This formula helps calculate the sum quickly and efficiently without having to add up each number individually. By applying this formula, we were able to solve the given problem with \( a = 9 \), \( l = 36 \), and \( n = 28 \).

• First Term (a): 9
• Last Term (l): 36
• Number of Terms (n): 28

Understanding this concept is crucial to solving many different types of problems involving sequences of numbers. It forms the foundation for more complex series and is frequently used in various mathematical applications.

Quadratic series involve the sum of squares of numbers. In mathematical terms, these series are represented as

• \( \sum_{k=1}^{n} k^2 \)

For part (b), the task was to evaluate the sum of squares starting from 3 to 17. This range needs a little adjustment before using the standard formula for the sum of squares:

• \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \)

To find the sum from a different starting point other than 1, like from 3 in this exercise, one approach is to subtract the sum of squares from 1 to 2 from the sum from 1 to 17.
Using this method allowed us to break down the calculation: first, determining the total sum of squares up to 17, then subtracting the sum from 1 to 2, which results in getting the sum exclusively from 3 to 17.
Calculating:

• \( \sum_{k=1}^{17} k^2 = 1785 \)
• \( \sum_{k=1}^{2} k^2 = 5 \)

This gives us the desired sum: \( 1785 - 5 = 1780 \). Understanding the sum of squares is vital in many mathematics and physics problems, particularly when dealing with the area, distance, and even in statistical formulas.

The sum of squares is a concept that often appears in mathematical series. It's important in our case for parts of series evaluation, especially when dealing with terms like \( k^2 \). For example, the exercise involving part (c) relies heavily on understanding this principle as it involves expressions such as \( k(k-1) = k^2 - k \).
In many cases, such expressions can be broken into simpler series:

• \( \sum_{k=a}^{b} k^2 \) - sum of squares
• \( \sum_{k=a}^{b} k \) - arithmetic series

Applying these principles allows the conversion of products into a combination of known series, which makes the calculation more straightforward. By using the formula for the sum of squares, namely:

• \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \)

we can derive the sum over any specified range. This same method helps solve our exercise for the specified values, resulting in the final sum by calculating the difference between two standard sums.
Utilizing such principles not only aids in bare calculations but also lays the ground for more complex mathematical operations and theories often used in advanced topics like calculus and statistical analyses.