Arithmetic Sequences $$ U_{40} = 2 + (40 - 1) \times 3 $$

Arithmetic Sequences

In this lesson, we study arithmetic sequences and learn how to find a specific term in the sequence. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.

1. Theory of Arithmetic Sequences

An arithmetic sequence follows a regular pattern by adding the same number, called the common difference, to each term. This type of sequence is widely used in mathematics and real-life situations such as calculating linear growth and financial planning.

2. Concept Explanation

To find any term in an arithmetic sequence, we need two key values: the first term and the common difference. Once these are known, we can use a formula to calculate the desired term directly, without listing all previous terms.

3. Steps or Formula

The formula for the nth term of an arithmetic sequence is:

$$ U_n = a + (n - 1)b $$

where:

• a is the first term
• b is the common difference
• n is the term number

4. Example Problem

Find the 40th term of the sequence:

2, 5, 8, 11, ...

Step 1: Identify known values.

$$ a = 2 $$ $$ b = 5 - 2 = 3 $$

Step 2: Substitute into the formula.

$$ U_{40} = 2 + (40 - 1) \times 3 $$

Step 3: Perform the calculation.

$$ U_{40} = 2 + 39 \times 3 $$ $$ U_{40} = 2 + 117 $$ $$ U_{40} = 119 $$

5. Final Answer

The 40th term of the arithmetic sequence is 119.

Arithmetic sequences are useful for modeling steady changes, such as simple interest, regular savings, and evenly increasing patterns. Understanding this concept makes it easier to analyze many real-world problems involving linear growth.