Problem 51 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. \( 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 \)

Step-by-Step Solution

Answer

The sum of the given arithmetic sequence is 110.

Step 1: Identify the sequence type

The given sequence follows a constant difference pattern, hence, the given sequence is an arithmetic sequence.

Step 2: Identify First and Last Term

In the given arithmetic sequence, the first term a1 is 2 and the last term an is 20.

Step 3: Identify the number of terms

The given arithmetic sequence has total terms 'n' from 2 to 20, with a common difference of 2. Thus there are 10 terms (2, 4, 6, 8, 10, 12, 14, 16, 18, 20).

Step 4: Apply the arithmetic sum formula

Apply arithmetic sum formula to calculate the sum of an arithmetic series. The formula for the sum 'S' of 'n' terms in an arithmetic sequence where 'a1' is the first term and 'an' is the last term is \(S = n/2 * (a1 + an)\). Substituting the values in this formula: \(S = 10/2 * (2 + 20)\).

Step 5: Calculate the sum

The sum of the arithmetic sequence S = 5 * 22 = 110

Key Concepts

Arithmetic Series

An arithmetic series is a sum of the terms of an arithmetic sequence. Whenever you see a sequence of numbers where the difference between each consecutive term is constant, you're dealing with an arithmetic sequence. When you take these numbers and add them together, it becomes an arithmetic series.

This concept is useful in various real-life situations, such as calculating the total number of seats in a theater with rows of seats that increase by a fixed number. In this exercise, if you're summing up numbers in order like 2, 4, 6, etc., and adding them all up, you're essentially finding the arithmetic series.

• An important part of working with an arithmetic series is identifying the first term and the last term of the sequence.
• These will guide how you apply the arithmetic sum formula to get your answer.

Common Difference

In an arithmetic sequence or series, the common difference is the amount added to each term to get the next term. It's a crucial component of understanding the pattern that connects the numbers in the sequence.

For example, in the sequence 2, 4, 6, 8, the common difference is 2, because each number increases by 2 from the previous number. Knowing the common difference helps you keep track of how many terms there are and identify the last term if it's not explicitly given.

• It also serves as a constant guide to maintaining the regular interval between numbers.
• The common difference can be calculated by subtracting any term from the subsequent term of the sequence.

Arithmetic Sum Formula

The arithmetic sum formula is a mathematical tool used to calculate the sum of an arithmetic series quickly and efficiently. Instead of adding each number one by one, you can use a formula to find the sum in a flash.

The formula is: \[ S = \frac{n}{2} \cdot (a_1 + a_n) \]Here,

• \( S \) represents the sum of the series,
• \( n \) is the total number of terms,
• \( a_1 \) is the first term,
• \( a_n \) is the last term.

Using our example from the exercise, the formula becomes \( S = \frac{10}{2} \cdot (2 + 20) \), which simplifies to \( S = 5 \cdot 22 = 110 \).

This formula is particularly helpful for students dealing with larger sequences, where counting manually would be impractical.