91Ó°ÊÓ

Sigma notation is a compact way to write down the sum of a sequence of terms. It is represented by the Greek capital letter sigma \( \( \Sigma \) \). The notation includes an expression for the terms of the sequence, and below and above the \( \Sigma \) symbol, you'll find the starting value and the ending value, respectively. This tells you the range of values to sum over. So, for example, when we see \( \sum_{i=1}^{4} \), it means we're going to add together the values for when \( i \) is 1, 2, 3, and 4. Such notation is particularly useful for long series, where writing out all the terms would be impractical.

In the given exercise, we have the sigma notation \( \sum_{i=1}^{4}\left[(i-1)^{2}+(i+1)^{3}\right] \). This means we calculate the given expression within the brackets for each integer value of \( i \) starting at 1 and ending at 4, then sum those values together.

The term \(sum of powers\) refers to the sum of terms raised to a certain power. When sequences involve terms with exponents, like in our exercise \( (i-1)^{2} \) and \( (i+1)^{3} \) for varying values of \( i \), we are dealing with a sum of powers. It is often helpful to work these out individually, especially when the sequence is short and the powers are relatively small.

Understanding how to expand such terms and then combine them for the sum is crucial. For example, for \( i = 1, (i-1)^{2} + (i+1)^{3} \) becomes \( 0^{2} + 2^{3} \), which simplifies to 0 plus 8, giving us the first term of the sum. Repeating this process for each value of \( i \) in our range and adding them gives us the total sum of the sequence.

An arithmetic sequence is a sequence of numbers where each term after the first is found by adding a constant to the previous term. This constant is called the common difference. While the given example is not an arithmetic sequence, because the terms do not have a consistent common difference, understanding arithmetic sequences is still useful. They are often simpler to sum than other sequences because there are formulae such as the familiar \( S_n = \frac{n}{2}(a_1 + a_n) \) that can find the sum of an arithmetic sequence quickly, where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

However, in cases like our exercise, where the sequence of terms does not have a constant difference, direct computation of each term using the provided formulas and summing them is the way to go. Summing up an arithmetic sequence efficiently often involves pairing up terms in a strategic way and is a skill worth developing for sequences that fit the arithmetic pattern.