Question

fi nd the sum. \(\sum_{i=1}^{33}(6-2 i)\)

Short answer

-924

Step-by-step solution

Identify the first term of the series a

Substitute i = 1 into the expression for the series. The first term a = \(6 - 2 \times 1 = 4\)

Identify the last term of the series l

Substitute i = 33 into the expression for the series. The last term l = \(6 - 2 \times 33 = -60\)

Calculate the number of terms n

Since we are summing from i = 1 to i = 33, the number of terms n = 33.

Substitute into the formula to find the sum

Substitute the values into the formula for the sum \(S = \frac{n}{2} (a + l) = \frac{33}{2} (4 - 60) = -924\)

Key concepts

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence, each term being generated by adding a constant called the common difference to the previous term. For example, in the series \( 2, 5, 8, 11, ... \) the common difference is \( 3 \).

To find the sum of an arithmetic series, \( S \), you can use the formula \( S = \frac{n}{2}(a + l) \) where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. This formula is derived under the premise that adding the first term to the last term, the second term to the second-to-last term, and so on, results in pairs that have the same total, making it easier to find the final sum.

Sigma Notation

Sigma notation, also known as summation notation, is a concise and convenient way to represent long sums. The notation uses the Greek letter Sigma (\(\Sigma\)) to indicate that a series of terms is to be added together. For example, the expression \(\sum_{i=1}^{n} a_i\) tells you to sum all the \( a_i \) terms from \( i = 1 \) to \( i = n \).

To break down sigma notation:

• The symbol \(\sum\) represents the summation.
• The variable below the \(\sum\), often \( i \) in this case, is called the index of summation.
• The lower bound (below the sigma) indicates the starting value of the index, and the upper bound (above the sigma) indicates the ending value.
• \(a_i\) represents the general term of the series, which depends on the index \(i\).

When solving exercises with sigma notation, it helps to work through the series element by element, especially when the general term is a function of the index, as in \(6 - 2i\).

Series and Sequences

Series and sequences are fundamental concepts in mathematics that involve ordered lists of numbers. While a sequence is simply the list of numbers in a particular order, a series is the sum of a sequence’s terms.

More formally:

• A sequence is a function from a subset of the integers, typically starting from 1, to a set of real numbers.
• A series is the summation of the elements of a sequence.

For the arithmetic sequence, the nth term can be found using the formula \(a_n = a + (n - 1)d\) where \( d \) is the common difference, and for the arithmetic series, the sum is found as mentioned earlier.

Understanding the difference between sequence and series is crucial for solving various kinds of problems in algebra and calculus. It allows students to analyze patterns and calculate sums over larger intervals without manually adding each term.