Arithmetic Sequence Word Problems- Solutions and Examples

What Are Arithmetic Sequence Word Problems?

Arithmetic sequence word problems are questions that describe a pattern where something changes by a constant amount each time. You have to find the next number, the nth term, or the sum of terms.

The trick? Most students fail because they try to memorize formulas instead of understanding what the problem is actually asking.

Here's the reality: if you can spot the pattern and apply two basic formulas, you can solve almost any arithmetic sequence problem.

The Two Formulas You Actually Need

Forget memorizing twenty different equations. You only need these two:

• nth term: a_n = a₁ + (n-1)d
• Sum of n terms: S_n = n/2 × (a₁ + a_n)

In these formulas:

• a₁ = first term
• d = common difference (what you add each time)
• n = term number you're looking for

How to Solve Any Arithmetic Sequence Word Problem

Here's the process that actually works:

1. Find the first term (a₁) — this is your starting point
2. Find the common difference (d) — subtract any term from the next term
3. Identify what the question wants — a specific term or a sum?
4. Plug into the right formula — use the table below to pick
5. Solve and check — verify your answer makes sense

Quick Reference: Which Formula Do I Use?

Example 1: The Paycheck Problem

Problem: You start a job earning $40,000 per year. You get a raise of $2,000 each year. What will your salary be in year 10?

Step 1: Identify a₁ and d

a₁ = 40,000 (year 1 salary)

d = 2,000 (raise amount)

Step 2: Apply the formula

a₁₀ = 40,000 + (10-1) × 2,000

a₁₀ = 40,000 + 9 × 2,000

a₁₀ = 40,000 + 18,000

Answer: $58,000

Example 2: The Seats in a Theater

Problem: A theater has 20 seats in the first row, 24 in the second row, and 28 in the third row. If this pattern continues, how many seats are in row 12?

Step 1: Find a₁ and d

a₁ = 20

d = 24 - 20 = 4

Step 2: Apply the formula

a₁₂ = 20 + (12-1) × 4

a₁₂ = 20 + 11 × 4

a₁₂ = 20 + 44

Answer: 64 seats

Example 3: Finding the Total (Sum Problem)

Problem: A savings account starts with $100 and increases by $50 each month. How much money is in the account after 12 months?

Step 1: Find a₁, d, and a₁₂

a₁ = 100

d = 50

a₁₂ = 100 + (12-1) × 50 = 100 + 550 = 650

Step 2: Use the sum formula

S₁₂ = 12/2 × (100 + 650)

S₁₂ = 6 × 750

Answer: $750 total saved

Example 4: When You Need to Find n

Problem: A tennis player serves 15 balls in the first practice session and increases by 3 each session. In which session will she serve 45 balls?

Set up the equation:

45 = 15 + (n-1) × 3

45 - 15 = (n-1) × 3

30 = (n-1) × 3

10 = n-1

Answer: Session 11

Common Mistakes That Cost You Points

• Forgetting to subtract 1 — the formula is (n-1)d, not nd
• Using the wrong d — always check by subtracting consecutive terms
• Mixed up sum formula — if you don't know a_n, use S_n = n/2 × [2a₁ + (n-1)d]
• Arithmetic errors — show your work, don't do it in your head

Getting Started: Your Action Plan

Before you tackle any problem:

1. Write down a₁ = ? and d = ? at the top of your work
2. Circle what the question actually wants (a specific term or a sum)
3. Pick your formula from the table above
4. Substitute the numbers carefully
5. Calculate — don't rush the arithmetic

Practice with 5 problems using this exact checklist. After that, the process becomes automatic.

When the Pattern Isn't Obvious

Some problems hide the arithmetic sequence. Look for phrases like:

• "increases by the same amount each time"
• "decreases by a constant rate"
• "adding the same number"
• "every month, you save $X more than the previous month"

These all describe arithmetic sequences. The wording changes, but the math stays the same.

One More Thing

If you're getting answers that don't make sense (like a negative number of people or negative money), check your common difference. You probably used the wrong value or subtracted in the wrong order.