Arithmetic Sequences Basics

Ever notice how some number patterns follow a simple "add the same amount each time" rule? That's an arithmetic sequence! In sequences like 2, 7, 12, 17, 22, ... each number increases by 5.

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This consistent difference is called the common difference (often written as "d").

To check if a sequence is arithmetic, just subtract each term from the one that follows it. If you always get the same answer, it's arithmetic! For example, in -5, -3, -1, 1, 3,...:

• $-3$ - $-5$ = 2
• $-1$ - $-3$ = 2
• 1 - $-1$ = 2
• 3 - 1 = 2

💡 The key to identifying arithmetic sequences is consistency! If the difference between consecutive terms stays the same throughout the entire sequence, you're looking at an arithmetic sequence.

Finding the nth Term

Once you know a sequence is arithmetic, you can find any term without writing out the whole sequence. That's super helpful when you need the 100th term!

The formula for finding the nth term is: an = a₁ + $n-1$d

• a₁ is your first term
• d is your common difference
• n is which term you're looking for

For example, with the sequence 57, 45, 33, 21,...:

1. First find d: 45 - 57 = -12 (each term decreases by 12)
2. Identify a₁ = 57
3. Use the formula: aₙ = 57 + n-1$$-12

To find the 19th term, just plug in n = 19: a₁₉ = 57 + 19-1$$-12 = 57 + 18$-12$ = 57 - 216 = -159

💡 Notice the pattern: the coefficient of d is always one less than the term number. This makes sense because you start with a₁ and then add d repeatedly.

Solving for Missing Information

You can use the arithmetic sequence formula to solve for any missing piece - whether that's the first term, the common difference, or a specific term number.

When you're given the nth term (aₙ) and the common difference $d$, you can find the first term (a₁):

1. Start with the formula: aₙ = a₁ + $n-1$d
2. Substitute what you know
3. Solve for a₁

For example, if a₁₁ = 41 and d = 5: 41 = a₁ + $11-1$(5) 41 = a₁ + 50 a₁ = -9

When given two different terms, you can set up a system:

• a₆ = 7 means a₁ + 5d = 7
• a₂₂ = 87 means a₁ + 21d = 87
• Solving these equations gives a₁ = -18 and d = 5

💡 Think of these problems as puzzles! You have a formula with multiple variables, and each piece of information helps you narrow down the missing values.

Arithmetic Series

What happens when you want to add up all the terms in an arithmetic sequence? That's an arithmetic series!

The difference between a sequence and a series:

• A sequence is a list of numbers (1, 3, 5, 7, 9,...)
• A series is the sum of those numbers $1 + 3 + 5 + 7 + 9 + ...$

Luckily, there's a formula to find the sum quickly: Sₙ = n/2$$a₁ + aₙ Where:

• n is the number of terms
• a₁ is the first term
• aₙ is the last term

For example, to find the sum of the first 15 terms where aₙ = 9 + 3n:

1. Find a₁ = 9 + 3(1) = 12
2. Find a₁₅ = 9 + 3(15) = 54
3. Apply the formula: S₁₅ = 15/2$$12 + 54 = $15/2$(66) = 495

💡 This formula saves you tons of time! Instead of adding 15 numbers, you only need to know the first term, last term, and how many terms you're adding.