Question

fi nd the sum. \(\sum_{i=1}^{31}(-3-4 i)\)

Short answer

-2082

Step-by-step solution

Identify the Elements in the Sum

The equation given to find the sum is \(\sum_{i=1}^{31}(-3-4 i)\). This series starts with \(i=1\) and ends with \(i=31\). The common difference here is \( -3-4i \).

Apply the Summation Formula

We can use the formula for arithmetic series sum to find the sum of this series. The formula is \(\frac{n}{2} \times (a+l)\), where n is the number of terms, a is the first term, and l is the last term. Here the number of terms n is 31. The first term (a) is obtained when \(i=1\), so \(a=-3-4(1)=-7\). The last term (l) is obtained when \(i=31\), so \(l=-3-4(31)=-127\). Plugging these values into the formula gives the sum.

Calculate the sum

Using the formula and above found values, we calculate \(\frac{31}{2} \times (-7-127)=-2082\).

Key concepts

Summation Notation

Summation notation is a compact way to represent the addition of series of numbers and is identified by the symbol \( \sum \). It allows us to sum up a sequence of numbers defined by a pattern, quickly and efficiently. For example, the expression \( \sum_{i=1}^{31}(-3-4i) \) tells us to sum up the numbers generated by the pattern \( -3-4i \) for each integer value of \( i \) as it goes from 1 to 31.

To understand this notation, we note that \( i \) is the variable, which in this case starts at 1 and ends at 31, which are the lower and upper bounds of our summation. The \( -3-4i \) part defines the sequence of numbers to be added or the 'terms' of the series. Each term is found by substituting the corresponding value of \( i \) into the pattern. In simple terms, with summation notation, we are adding up a bunch of terms quickly, instead of writing out the long repetitive process of adding each term individually.

Series and Sequences

Series and sequences are fundamental concepts in mathematics, especially when dealing with lists of numbers. A sequence is an ordered list of numbers that often follow a specific rule, like the sequence of even numbers \(2, 4, 6, 8, \dots \). A series, on the other hand, is what we get when we add up the terms of a sequence.

Sequences can be finite or infinite, depending on whether they have an end or not. Series can also take these forms. In this context, when we refer to the series \( \sum_{i=1}^{31}(-3-4i) \) from the exercise, we are dealing with a finite series because it has a clearly defined first term and a last term. Understanding the relationship between series and sequences is crucial for solving problems, as the series represents the sum of all terms of the sequence within given bounds.

Arithmetic Series Formula

The arithmetic series formula is a tool that helps us to find the sum of an arithmetic series. An arithmetic series is the sum of terms of an arithmetic sequence, which is a sequence where each term after the first is obtained by adding a constant, known as the common difference, to the previous term.

The formula for the sum of an arithmetic series is \( \frac{n}{2} \times (a+l) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term of the series. This formula applies when the common difference is consistent for all terms. In the given exercise, we applied this formula by identifying the first term \( a = -7 \) when \( i = 1 \), and the last term \( l = -127 \) when \( i = 31 \). With these values and the number of terms \( n = 31 \) plugged in, we used the arithmetic series formula to find the sum, \( -2082 \) efficiently without having to add each term one by one.