To understand an arithmetic sequence and solve related problems step-by-step, let's break down the process.

Definition of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference ($d$), to the previous term.

The general form of an arithmetic sequence is: $a, \ a + d, \ a + 2d, \ a + 3d, \ \ldots$ where:

• $a$ is the first term.
• $d$ is the common difference.

Step-by-Step Guide

Step 1: Identify the First Term ($a$)

• This is the first number in your sequence. Let’s denote it by $a$.

Step 2: Determine the Common Difference ($d$)

• Find the difference between any two consecutive terms. If the sequence is arithmetic, this difference will remain constant.

  $d = \text{Second Term} - \text{First Term}$

Step 3: Use the Formula for the $n$-th Term

The formula for the $n$-th term ($a_n$) of an arithmetic sequence is: $a_n = a + (n - 1)d$ where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number you want to find.

Step 4: Calculate the Sum of the First $n$ Terms

The sum of the first $n$ terms ($S_n$) of an arithmetic sequence is given by: $S_n = \frac{n}{2} \left( 2a + (n - 1)d \right)$ or $S_n = \frac{n}{2} \left( a + a_n \right)$ This formula is useful when you want to find the sum up to a specific term.

Example Problem

Let's say we have an arithmetic sequence: $3, 7, 11, 15, \ldots$, and we want to find the 10th term and the sum of the first 10 terms.

Step-by-Step Solution

1. Identify the First Term ($a$):

   • Here, $a = 3$.

2. Determine the Common Difference ($d$):

   • $d = 7 - 3 = 4$.

3. Find the 10th Term ($a_{10}$):

   • Using the formula $a_n = a + (n - 1)d$: $a_{10} = 3 + (10 - 1) \cdot 4$ $a_{10} = 3 + 9 \cdot 4$ $a_{10} = 3 + 36 = 39$

4. Calculate the Sum of the First 10 Terms ($S_{10}$):

   • Using the formula $S_n = \frac{n}{2} \left(2a + (n - 1)d\right)$: $S_{10} = \frac{10}{2} \left(2 \cdot 3 + (10 - 1) \cdot 4\right)$ $S_{10} = 5 \left(6 + 36\right)$ $S_{10} = 5 \cdot 42 = 210$

Summary

• 10th term: $39$
• Sum of the first 10 terms: $210$

Would you like further details or have any questions?

Here are some related questions:

1. What if the common difference in an arithmetic sequence is negative?
2. How do you determine if a sequence is arithmetic?
3. Can arithmetic sequences be applied to real-life scenarios?
4. How would you find the sum of an infinite arithmetic sequence?
5. What is the difference between arithmetic and geometric sequences?

Tip: When identifying a sequence, always check for consistency in the common difference to confirm it's arithmetic.