Arithmetic Sequences: Recognize Series

Grade 12 · algebra · 83 practice problems · read aloud

🔊 Listen to this explanation

📈 Arithmetic Sequences: The Linear Pattern

An arithmetic sequence is an ordered list of numbers where the difference between consecutive terms is always the same. This constant difference is called the common difference (d). Understanding them is crucial for modeling linear growth in finance, computer science, and physics.

🔧 The Core Formula

The explicit (nth term) formula for any arithmetic sequence is:

aₙ = a₁ + (n - 1)d

  • aₙ is the nth term you're finding.
  • a₁ is the first term in the sequence.
  • n is the term number (e.g., 1st, 2nd, 10th).
  • d is the common difference (a₂ - a₁).

🧮 Step-by-Step Guide

  1. Identify a₁ and d from the sequence.
  2. Plug the values into the formula: aₙ = a₁ + (n - 1)d.
  3. Simplify the expression to solve for the desired term.

✨ Worked Examples

Example 1: Find the 15th term of the sequence 7, 11, 15, 19, ...

  1. a₁ = 7, d = 11 - 7 = 4
  2. a₁₅ = 7 + (15 - 1) * 4
  3. a₁₅ = 7 + (14) * 4
  4. a₁₅ = 7 + 56 = 63

Example 2: A sequence has a₁ = -5 and d = 3. Write the explicit formula and find the 20th term.

  1. Formula: aₙ = -5 + (n - 1) * 3
  2. Simplify: aₙ = -5 + 3n - 3 → aₙ = 3n - 8
  3. a₂₀ = 3(20) - 8 = 60 - 8 = 52

⚠️ Common Mistakes

  • Using "n" instead of "(n-1)": The multiplier for 'd' is the number of jumps, not the term number. For the 10th term, you make 9 jumps of size 'd'.
  • Misidentifying 'd': Always subtract a term from the one after it (a₂ - a₁). A negative common difference means the sequence is decreasing.
  • Incorrect Order of Operations: Remember PEMDAS. Multiply (n-1) by 'd' before adding to a₁.

💡 Tips & Tricks

  • Memory Aid: Think "First term + (Jumps) * (Difference)".
  • Quick Check: Your formula aₙ should produce the known first term when n=1.
  • Graphical Link: The points (n, aₙ) form a straight line with a slope equal to 'd'.

🏋️ Practice Suggestions

  • Start by finding the common difference in various sequences (increasing and decreasing).
  • Practice writing the explicit formula from the first term and common difference.
  • Solve mixed problems: find a specific term, find the term number given its value, or find the first term given a later term and the common difference.
  • Create your own sequences and solve for terms to build deep understanding.

Practice problems

6 of the 83, worked through step by step — try them before opening the answer.

1 Arithmetic sequence: a₁=11, d=8. Write explicit formula

Hint: The explicit formula for an arithmetic sequence uses the first term and common difference to find any term directly

Show the answer

Answer: aₙ = 11 + (n-1)8

  1. Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d
  2. Substitute the given values: a₁ = 11 and d = 8
  3. The formula becomes: aₙ = 11 + (n-1)8
  4. This is the explicit formula that can be used to find any term in the sequence The explicit formula is aₙ = 11 + (n-1)8

2 Arithmetic sequence: a₁=8, d=5. Write explicit formula aₙ = ?

Hint: The explicit formula for an arithmetic sequence uses the first term and common difference to express any term directly

Show the answer

Answer: aₙ = 5n + 3

  1. The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d
  2. Substitute the given values: a₁ = 8 and d = 5
  3. aₙ = 8 + (n-1)5
  4. Distribute the 5: aₙ = 8 + 5n - 5
  5. Combine like terms: aₙ = 5n + 3 The explicit formula is aₙ = 5n + 3

3 Arithmetic sequence: a₁=6, d=4. Write explicit formula aₙ = ?

Hint: The explicit formula for an arithmetic sequence uses the first term and common difference in a linear pattern

Show the answer

Answer: aₙ = 4n + 2

  1. The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d
  2. Substitute the given values: aₙ = 6 + (n-1)×4
  3. Distribute the 4: aₙ = 6 + 4n - 4
  4. Combine like terms: aₙ = 4n + 2 The explicit formula is aₙ = 4n + 2

4 Arithmetic sequence: a₁=9, d=7. Write explicit formula aₙ = ?

Hint: The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d. Apply this formula with the given values.

Show the answer

Answer: aₙ = 7n + 2

  1. Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d
  2. Substitute the given values: a₁ = 9 and d = 7
  3. aₙ = 9 + (n-1)×7
  4. Distribute the 7: aₙ = 9 + 7n - 7
  5. Combine like terms: aₙ = 7n + 2
  6. The explicit formula is aₙ = 7n + 2

5 Arithmetic sequence: a₁=7, d=9. Write explicit formula aₙ = ?

Hint: Remember the general formula for an arithmetic sequence is aₙ = a₁ + (n-1)d. Substitute the given values and simplify.

Show the answer

Answer: aₙ = 9n - 2

  1. Start with the general formula for an arithmetic sequence: aₙ = a₁ + (n-1)d
  2. Substitute the given values: a₁ = 7 and d = 9
  3. aₙ = 7 + (n-1)×9
  4. Distribute the 9: aₙ = 7 + 9n - 9
  5. Combine like terms: aₙ = 9n - 2 The explicit formula is aₙ = 9n - 2.

6 Arithmetic sequence: a₁=11, d=5. Write explicit formula aₙ = ?

Hint: The explicit formula for an arithmetic sequence uses the first term and common difference to find any term

Show the answer

Answer: aₙ = 11 + (n-1)5

  1. Recall the explicit formula for an arithmetic sequence: aₙ = a₁ + (n-1)d
  2. Substitute the given values: a₁ = 11 and d = 5
  3. aₙ = 11 + (n-1)5
  4. The explicit formula is aₙ = 11 + (n-1)5
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