11.2 Arithmetic Sequences

Arithmetic sequences are number patterns where each term differs from the previous one by a constant amount called the common difference. This constant difference is what makes arithmetic sequences predictable and connects them directly to linear functions you've already studied.

You can describe arithmetic sequences using recursive or explicit formulas. Recursive formulas define each term based on the previous one, while explicit formulas let you jump straight to any term. Both show up frequently in problems involving linear growth, financial planning, and pattern recognition.

Arithmetic Sequences

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Common difference in arithmetic sequences

The common difference ($d$) is the constant value added to each term to produce the next term. You find it by subtracting any term from the term that follows it:

$d = a_{n+1} - a_n$

This difference stays the same between every pair of consecutive terms. If it doesn't, the sequence isn't arithmetic.

• A positive $d$ means the sequence is increasing
• A negative $d$ means the sequence is decreasing
• A $d$ of zero means every term is the same (constant sequence)

Example: In the sequence 2, 5, 8, 11, 14, the common difference is 3. You can verify this across every pair: $5 - 2 = 3$, $8 - 5 = 3$, $11 - 8 = 3$, $14 - 11 = 3$.

Recursive formulas for arithmetic sequences

A recursive formula defines each term using the term right before it. The general form is:

$a_{n+1} = a_n + d$

where $a_n$ is the current term and $d$ is the common difference. To use a recursive formula, you always need two pieces of information: the first term ($a_1$) and the common difference ($d$).

Example: For the sequence 3, 7, 11, 15, 19, the recursive formula is $a_{n+1} = a_n + 4$ with $a_1 = 3$. To find the 4th term, you'd need to calculate the 2nd and 3rd terms first. That's the main limitation of recursive formulas: you have to work through every preceding term to reach the one you want.

Explicit formulas for arithmetic sequences

The explicit formula lets you find any term directly without computing all the terms before it:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term's position number, and $d$ is the common difference.

Example: For the sequence 5, 9, 13, 17, 21, you have $a_1 = 5$ and $d = 4$, so the explicit formula is $a_n = 5 + (n - 1)(4)$. Want the 50th term? Just plug in $n = 50$:

$a_{50} = 5 + (50 - 1)(4) = 5 + 196 = 201$

Notice that this formula has the same structure as a linear equation. If you distribute and simplify $a_n = 5 + 4(n-1) = 4n + 1$, you can see it's equivalent to $y = mx + b$ where the common difference $d$ acts as the slope and the y-intercept relates to $a_1$. This is why arithmetic sequences always produce a straight line when you plot term number vs. term value.

Real-world applications of arithmetic sequences

Arithmetic sequences model any situation where a quantity increases or decreases by a fixed amount at regular intervals:

• Financial planning: Saving an extra $50 each month on top of a base deposit follows an arithmetic pattern
• Seating arrangements: An auditorium where each row has a fixed number of additional seats compared to the row in front of it
• Depreciation: Equipment that loses the same dollar amount of value each year

Example: A concert hall has 20 seats in the first row, and each subsequent row has 2 more seats than the previous row. Here $a_1 = 20$ and $d = 2$. To find how many seats are in the 15th row: $a_{15} = 20 + (15 - 1)(2) = 20 + 28 = 48$ seats.

Solving problems with arithmetic sequences

1. Identify the first term ($a_1$) and the common difference ($d$) from the information given
2. Choose the appropriate formula: use recursive if you're building term-by-term, or explicit if you need a specific term far into the sequence
3. Substitute the known values into the formula
4. Solve for the unknown variable and check that your answer makes sense in context

A common twist: sometimes you're given two non-consecutive terms and asked to find $d$ or $a_1$. Set up the explicit formula for each given term and solve the resulting system.

Additional concepts in arithmetic sequences

• Sequence: An ordered list of numbers following a specific pattern
• Term: Each individual number in a sequence, identified by its position ($a_1, a_2, a_3, \ldots$)
• Arithmetic mean: The average of two terms in an arithmetic sequence equals the term exactly halfway between them. For example, in the sequence 4, 7, 10, the arithmetic mean of 4 and 10 is $\frac{4 + 10}{2} = 7$, which is the middle term.
• Summation: Adding up all terms in a sequence. For arithmetic sequences specifically, you'll use the formula $S_n = \frac{n}{2}(a_1 + a_n)$, which comes up in the next section on series.