Question

Write the sum using sigma notation. $$ 40+35+30+25+20+15+10+5 $$

Short answer

Question: Express the sum 40 + 35 + 30 + 25 + 20 + 15 + 10 + 5 using sigma notation. Answer: The given sum can be expressed in sigma notation as: $$ \sum_{n=1}^{8} (40 + (n - 1) (-5)) $$

Step-by-step solution

Identify the pattern

First, let's look at the given sum: $$ 40 + 35 + 30 + 25 + 20 + 15 + 10 + 5 $$ We can see that each term in the sum is 5 less than the previous term. In other words, the terms form an arithmetic sequence with a common difference of -5.

Determine the initial value and final value of the index

In order to represent the sum in sigma notation, we need to identify the initial value (n_initial) and final value (n_final) of the index, n. Let's start with the initial value of n, which corresponds to the first term, 40. We can represent the initial value as: $$ n_{initial} = 1 $$ Now, we need to find the final value of the index, n, that corresponds to the last term, 5. Since there are 8 terms in the sum, we can represent the final value of the index as: $$ n_{final} = 8 $$

Write the general term of the arithmetic sequence

To write the sum in sigma notation, we need to express the general term (an) of the arithmetic sequence. Given that the common difference (d) is -5 and the first term (a1) is 40, we can write the general term as: $$ a_n = a_1 + (n - 1) d = 40 + (n - 1) (-5) $$

Write the sum using sigma notation

Now that we have determined the initial and final values of the index and the general term of the arithmetic sequence, we can write the sum using sigma notation: $$ \sum_{n=1}^{8} (40 + (n - 1) (-5)) $$ So, the given sum can be expressed in sigma notation as: $$ \sum_{n=1}^{8} (40 + (n - 1) (-5)) $$

Key concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value to the previous term. This constant value is known as the 'common difference'.

In the sequence from the problem (i.e., 40, 35, 30, 25, 20, 15, 10, 5), each term is derived by subtracting 5 from the previous one. That makes it an arithmetic sequence with a common difference of -5.

• The first term of this sequence is 40.
• There are eight carefully structured terms.
• Once the pattern is identified, it can be used to describe the entire sequence and find other characteristics easily, such as the sum or additional terms.

Understanding the structure of an arithmetic sequence is fundamental when seeking solutions involving these sequences and is key to simplifying complex mathematical operations.

Common Difference

The common difference in an arithmetic sequence is the fixed amount added to or subtracted from one term to get to the next term. It is a constant and is the same for each pair of consecutive terms throughout the sequence.

In our example, to determine the common difference, simply calculate how much one term differs from the next. Here, that difference is \(-5\). Each number decreases by 5 from the previous one, which confirms our common difference is\(-5\).

• When identified, the common difference enables prediction of future terms.
• It also aids in writing the sequence formula or determining the completion of an arithmetic series.

Whether you're moving forward to predict new numbers or looking backward to understand prior terms, the common difference is an essential element in navigating arithmetic sequences.

General Term

In arithmetic sequences, the general term offers a formulaic way to find any term in the sequence without listing all terms preceding it. This formula uses the first term and the common difference. It is vital for expressing the sequence efficiently and finding specific numbers at different positions, especially for large sequences.
Given the steps to writing the general term, formula: \[ a_n = a_1 + (n - 1)d \]where -\( a_n \) represents the general term you want to find -\( a_1 \) is the first term -\( n \) is the position of the term - \( d \) is the common difference.

Applying to the provided example, the first term \( a_1 \) is 40, and the common difference \( d \) is -5. This derives the general formula: \[ a_n = 40 + (n - 1)(-5) \]

• With this tool, you can calculate any term in the sequence directly.
• It's particularly useful for re-writing sums using sigma notation and for quickly analyzing and manipulating sequences.