Arithmetic Sequences | CSE Reviewer

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where you add (or subtract) the same number each time to get the next number.

For example: 3, 7, 11, 15, 19...

Here, you add 4 each time:

• 3 + 4 = 7
• 7 + 4 = 11
• 11 + 4 = 15
• 15 + 4 = 19

Tip: The key word is "same." You always add the same number in an arithmetic sequence.

Example: Is 5, 10, 15, 20 an arithmetic sequence?

Step 1: Check the difference between numbers
- 10 - 5 = 5
- 15 - 10 = 5
- 20 - 15 = 5
Step 2: All differences are the same (5)

Answer: YES, it is an arithmetic sequence

Meaning: An arithmetic sequence adds (or subtracts) the same number every time to get the next term.

Key Parts of an Arithmetic Sequence

There are 3 important parts in every arithmetic sequence:

• a₁ (first term): The first number in the sequence
• d (common difference): The number you add each time
• n (term number): Which position you want (1st, 2nd, 3rd, etc.)

Example: In the sequence 3, 7, 11, 15, 19...

- a₁ = 3 (first term)
- d = 4 (common difference: 7 - 3 = 4)
- If you want the 5th term, n = 5

Example 1: Find a₁ and d in 10, 15, 20, 25...

Step 1: a₁ = 10 (first number)
Step 2: d = 15 - 10 = 5

Answer: a₁ = 10, d = 5

Example 2: Find a₁ and d in 50, 45, 40, 35...

Step 1: a₁ = 50
Step 2: d = 45 - 50 = -5 (negative because it decreases)

Answer: a₁ = 50, d = -5

Meaning: Every arithmetic sequence has a first term (a₁), a common difference (d), and term numbers (n).

The Formula for Any Term

To find any term in an arithmetic sequence, use this formula:

aₙ = a₁ + (n - 1)d

This means:

• aₙ = the term you want to find
• a₁ = first term
• n = which term number (1st, 2nd, 3rd, etc.)
• d = common difference

Shortcut: Think: aₙ = start + (how many times you add d)

Example 1: Find the 10th term. a₁ = 3, d = 4, n = 10

Step 1: Write formula → aₙ = a₁ + (n - 1)d
Step 2: Plug in → a₁₀ = 3 + (10 - 1) × 4
Step 3: Calculate → a₁₀ = 3 + 9 × 4
Step 4: a₁₀ = 3 + 36 = 39

Answer: The 10th term is 39

Example 2: Find the 15th term. Sequence: 5, 9, 13, 17...

Step 1: Find a₁ → a₁ = 5
Step 2: Find d → d = 9 - 5 = 4
Step 3: n = 15
Step 4: Plug into formula → a₁₅ = 5 + (15 - 1) × 4
Step 5: a₁₅ = 5 + 14 × 4
Step 6: a₁₅ = 5 + 56 = 61

Answer: The 15th term is 61

Example 3: Find the 8th term. Sequence: 20, 17, 14, 11...

Step 1: a₁ = 20
Step 2: d = 17 - 20 = -3
Step 3: n = 8
Step 4: a₈ = 20 + (8 - 1) × (-3)
Step 5: a₈ = 20 + 7 × (-3)
Step 6: a₈ = 20 + (-21) = -1

Answer: The 8th term is -1

Meaning: Use aₙ = a₁ + (n - 1)d to find any term. Plug in a₁, d, and n, then calculate.

Finding the Common Difference (d)

To find d, subtract any term from the term that comes after it.

Formula: d = a₂ - a₁ (or any two consecutive terms)

Tip: Always subtract the smaller term from the larger term (the one after it).

Example 1: Find d in 7, 12, 17, 22...

Step 1: d = 12 - 7
Step 2: d = 5

Answer: d = 5

Example 2: Find d in 30, 25, 20, 15...

Step 1: d = 25 - 30
Step 2: d = -5 (negative because it's decreasing)

Answer: d = -5

Meaning: To find d, subtract one term from the next term: d = next - current.

Finding the First Term (a₁)

If you know a later term and d, you can find a₁ by working backwards.

Use the formula and solve for a₁:

a₁ = aₙ - (n - 1)d

Example: The 10th term is 53. d = 5. Find a₁.

Step 1: Write formula → a₁ = aₙ - (n - 1)d
Step 2: Plug in → a₁ = 53 - (10 - 1) × 5
Step 3: a₁ = 53 - 9 × 5
Step 4: a₁ = 53 - 45 = 8

Answer: The first term is 8

Meaning: To find a₁, subtract (n - 1)d from the known term.

Finding n (Which Term Number)

If you know a term value, a₁, and d, you can find n by solving the formula.

Example: In the sequence 3, 7, 11, 15..., which term is 59?

Step 1: a₁ = 3, d = 4, aₙ = 59
Step 2: Write formula → 59 = 3 + (n - 1) × 4
Step 3: Solve for n:
59 = 3 + 4(n - 1)
59 - 3 = 4(n - 1)
56 = 4(n - 1)
56 ÷ 4 = n - 1
14 = n - 1
n = 15

Answer: 59 is the 15th term

Meaning: Plug in aₙ, a₁, and d, then solve the equation for n.

Tips and Shortcuts for Arithmetic Sequences

Arithmetic sequences can be fast if you use shortcuts. Here are the best tips.

Tips:

• Always find a₁ and d first before using the formula.
• d = next term - current term (use any two consecutive terms).
• If the sequence increases, d is positive.
• If the sequence decreases, d is negative.
• aₙ = a₁ + (n - 1)d is the main formula.
• For (n - 1), subtract 1 from n first, then multiply by d.
• Parentheses matter: (n - 1) × d, not n - 1 × d.

Example Shortcut: When counting terms, remember: (last - first) ÷ d + 1 = number of terms.

Example: How many numbers from 5 to 50 if you add 5 each time?
(50 - 5) ÷ 5 + 1 = 45 ÷ 5 + 1 = 9 + 1 = 10 terms

More Practice Examples

Example 1: Find the 20th term. a₁ = 2, d = 3
Step 1: a₂₀ = 2 + (20 - 1) × 3
Step 2: a₂₀ = 2 + 19 × 3
Step 3: a₂₀ = 2 + 57 = 59
Answer: 59

Example 2: Find d. Sequence: 12, 18, 24, 30...
Step 1: d = 18 - 12
Step 2: d = 6
Answer: d = 6

Example 3: Find the 12th term. Sequence: 100, 95, 90, 85...
Step 1: a₁ = 100
Step 2: d = 95 - 100 = -5
Step 3: a₁₂ = 100 + (12 - 1) × (-5)
Step 4: a₁₂ = 100 + 11 × (-5)
Step 5: a₁₂ = 100 - 55 = 45
Answer: 45

Example 4: Which term is 100? Sequence: 5, 9, 13, 17...
Step 1: a₁ = 5, d = 4, aₙ = 100
Step 2: 100 = 5 + (n - 1) × 4
Step 3: 95 = 4(n - 1)
Step 4: 95 ÷ 4 = 23.75 → Not a whole number
Answer: 100 is NOT in this sequence

Example 5: Find a₁. The 8th term is 45. d = 6
Step 1: a₁ = 45 - (8 - 1) × 6
Step 2: a₁ = 45 - 7 × 6
Step 3: a₁ = 45 - 42 = 3
Answer: a₁ = 3

What To Remember

• Arithmetic Sequence - A list where you add the same number each time.
• a₁ (First Term) - The first number in the sequence.
• d (Common Difference) - The number you add each time.
• n (Term Number) - Which position (1st, 2nd, 3rd, etc.).
• aₙ (Any Term) - The term you want to find.
• Main Formula - aₙ = a₁ + (n - 1)d
• Find d - d = next term - current term
• Find a₁ - a₁ = aₙ - (n - 1)d
• Find n - Solve aₙ = a₁ + (n - 1)d for n
• Positive d - Sequence increases
• Negative d - Sequence decreases
• Parentheses - (n - 1) × d, not n - 1 × d

Multiple Choice Questions

Answer these in the comment section. Choose the best answer for each item.

1. Find the 10th term. Sequence: 2, 5, 8, 11...

A. 25
B. 27
C. 29
D. 31

2. What is the common difference (d)? Sequence: 20, 15, 10, 5...

A. 5
B. -5
C. 10
D. -10

3. Find the 15th term. a₁ = 10, d = 3

A. 45
B. 50
C. 55
D. 60

4. Which term is 47? Sequence: 3, 7, 11, 15...

A. 10th
B. 11th
C. 12th
D. 13th

5. Find a₁. The 6th term is 32. d = 5

A. 5
B. 7
C. 9
D. 11