Finding the Common Difference

Find the common difference for the sequence: $8, 5, 2, -1, -4, \ldots$

Choose two consecutive terms: Let's use $a_1 = 8$ and $a_2 = 5$ = First two terms identified

Subtract the first from the second: $d = a_2 - a_1 = 5 - 8 = -3$ = $d = -3$

Verify with another pair: $2 - 5 = -3$ and $-1 - 2 = -3$ = Confirmed: $d = -3$

Answer: The common difference is $d = -3$. This is a decreasing arithmetic sequence.

Using the Explicit Formula

Find the 25th term of the arithmetic sequence: $7, 12, 17, 22, \ldots$

Identify $a_1$ and $d$: $a_1 = 7$, $d = 12 - 7 = 5$ = $a_1 = 7$, $d = 5$

Write the explicit formula: $a_n = a_1 + (n - 1) \cdot d$ = $a_n = 7 + (n - 1) \cdot 5$

Substitute $n = 25$: $a_{25} = 7 + (25 - 1) \cdot 5$ = $a_{25} = 7 + 24 \cdot 5$

Calculate: $a_{25} = 7 + 120 = 127$ = $a_{25} = 127$

Answer: The 25th term is $a_{25} = 127$

Finding a Term Given Two Terms

In an arithmetic sequence, $a_5 = 23$ and $a_{12} = 51$. Find $a_1$ and $d$.

Set up equations using the formula: $a_5 = a_1 + 4d = 23$ $a_{12} = a_1 + 11d = 51$ = Two equations with two unknowns

Subtract the first equation from the second: $(a_1 + 11d) - (a_1 + 4d) = 51 - 23$ $7d = 28$ = $7d = 28$

Solve for $d$: $d = 28 \div 7 = 4$ = $d = 4$

Substitute $d$ back to find $a_1$: $a_1 + 4(4) = 23$ $a_1 + 16 = 23$ $a_1 = 7$ = $a_1 = 7$

Answer: The first term is $a_1 = 7$ and the common difference is $d = 4$. The sequence is $7, 11, 15, 19, 23, \ldots$

Real-World Application

A concert hall has 20 seats in the first row and each subsequent row has 3 more seats than the row in front. How many seats are in the 15th row?

Identify the sequence parameters: $a_1 = 20$ (first row), $d = 3$ (increase per row) = $a_1 = 20$, $d = 3$

Write the explicit formula: $a_n = 20 + (n - 1) \cdot 3$ = $a_n = 20 + 3n - 3 = 17 + 3n$

Find $a_{15}$: $a_{15} = 20 + (15 - 1) \cdot 3$ $a_{15} = 20 + 14 \cdot 3$ $a_{15} = 20 + 42$ = $a_{15} = 62$

Answer: The 15th row has 62 seats.

Mistake: Using $n$ instead of $(n-1)$ in the formula

Why: The formula $a_n = a_1 + nd$ would give you the $(n+1)$th term, not the $n$th term. When $n=1$, we should get $a_1$, which requires $(n-1) = 0$.

Correct: Always use $a_n = a_1 + (n-1) \cdot d$. Check: when $n=1$, $a_1 = a_1 + 0 \cdot d = a_1$.

Mistake: Subtracting in the wrong order when finding $d$

Why: Students sometimes calculate $a_n - a_{n+1}$ instead of $a_{n+1} - a_n$, which gives the wrong sign for $d$.

Correct: Always subtract the earlier term from the later term: $d = a_{n+1} - a_n$

Mistake: Forgetting that $d$ can be negative

Why: Arithmetic sequences can decrease! A sequence like $20, 17, 14, 11, \ldots$ has $d = -3$.

Correct: Check if the sequence is increasing ($d > 0$) or decreasing ($d < 0$) before writing the formula.

Mistake: Confusing the term number with the term value

Why: In $a_5 = 23$, the subscript 5 is the position (5th term), and 23 is the value.

Correct: $a_n$ means the term at position $n$. The subscript is always the position.

Savings and Financial Planning

Regular savings deposits create arithmetic sequences. Understanding these patterns helps with financial planning.

If you save 75 euros per month starting from 0, after $n$ months you have $75n$ euros. If you start with 200 euros, your balance forms the sequence $200, 275, 350, 425, \ldots$

Construction and Architecture

Builders use arithmetic sequences when planning staircases, seating, and structural patterns.

A staircase has a total rise of 3 meters over 15 steps. Each step rises by $\frac{3}{15} = 0.2$ meters. The heights form the sequence $0.2, 0.4, 0.6, \ldots, 3.0$ meters.

Sports and Fitness

Training programs often increase intensity following arithmetic patterns.

A runner starts with 2 km and adds 0.5 km each week. After $n$ weeks, they run $2 + 0.5(n-1)$ km. Week 10: $2 + 0.5 \times 9 = 6.5$ km.

Q: What is the difference between an arithmetic sequence and an arithmetic series?

A: A sequence is a list of numbers following a pattern (e.g., $2, 5, 8, 11, \ldots$). A series is the sum of the terms in a sequence (e.g., $2 + 5 + 8 + 11 + \ldots$). We study arithmetic series after mastering sequences.

Q: Can the common difference be zero?

A: Yes! If $d = 0$, every term is the same: $5, 5, 5, 5, \ldots$ is technically an arithmetic sequence with $d = 0$.

Q: How do I know if a sequence is arithmetic?

A: Calculate the differences between consecutive terms. If all differences are equal, it is arithmetic. For $3, 7, 11, 15$: $7-3=4$, $11-7=4$, $15-11=4$. All differences are 4, so it is arithmetic with $d=4$.

Q: What if I am given two non-consecutive terms?

A: Use the formula to set up equations. If $a_3 = 10$ and $a_7 = 22$, then $a_1 + 2d = 10$ and $a_1 + 6d = 22$. Subtract to get $4d = 12$, so $d = 3$.