Sequence Classification: Differences & Ratios

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Part 1: Test the Pattern

Reminder

Arithmetic (linear): constant first difference. Geometric (exponential): constant common ratio.

First differences: subtract consecutive terms. Second differences: find the differences of the first differences.

A constant second difference suggests a quadratic sequence. A short sequence may fit more than one possible rule.

1. Complete the first differences and classify the sequence.

Sequence: 8, 12, 16, 20, ______, ______

First differences: ______, ______, ______, ______, ______

Classification: ____________________ Rule: T_n = ____________________

2. Complete the first and second differences, then classify the sequence.

Sequence: 2, 5, 10, 17, ______, ______

First differences: ______, ______, ______, ______, ______

Second differences: ______, ______, ______, ______

Classification: ____________________ Rule: T_n = ____________________

3. Calculate the common ratios and classify the sequence.

Sequence: 81, 27, 9, 3, ______, ______

Common ratios: ______, ______, ______, ______, ______

Classification: ____________________ Rule: T_n = ____________________

4. Calculate the common ratio and continue the sequence by three terms.

Sequence: 1.5, 0.75, 0.375, ______, ______, ______

Common ratio: ____________________ Classification: ____________________

Part 2: Classify and Justify

For each sequence, calculate the most useful evidence. Write linear, quadratic, exponential, or not enough information. Then complete the sentence.

5. Sequence A: 5, 9, 13, 17, ______, ______

Useful test and results: __________________________________________________________

It is ____________________ because __________________________________________________.

6. Sequence B: 1, 4, 9, 16, ______, ______

First differences: ____________________ Second differences: ____________________

It is ____________________ because __________________________________________________.

7. Sequence C: 5, 10, 20, 40, ______, ______

Common ratio: ____________________ It is ____________________ because _______________.

Rule: T_n = ____________________

8. Sequence D: 20, 17, 14, 11, ______, ______

First difference: ____________________ It is ____________________ because _____________.

9. Sequence E: 3, 7, 13, 21, ______, ______

First differences: ____________________ Second differences: ____________________

It is ____________________ because __________________________________________________.

10. Mixed reasoning: For the sequence 2, 4, 6, 8, explain why the next term is not guaranteed to be 10. Give one other possible next term and state what extra information would help.

Part 3: Challenges, Self-Check and Answer Key

11. Challenge: A sequence is 6, 10, ______, 18, 22. Find the missing term and justify your answer. Then write a possible nth-term rule.

12. Challenge: Find a possible nth-term rule for 4, 9, 16, 25. Check your rule for n = 1 and n = 4.

Self-check

☐ I can calculate first differences accurately.

☐ I can calculate second differences when first differences are not constant.

☐ I can test for a constant ratio, including fractions, decimals and negative values.

☐ I can classify a sequence and justify my decision using mathematical vocabulary.

☐ I can check an nth-term rule by substituting n = 1 and other positions.

☐ I understand that a short sequence may not have one unique rule.

Answer Key

1. Terms: 24, 28. First differences: +4 throughout. Linear; T_n = 4n + 4.

2. Terms: 26, 37. First differences: 3, 5, 7, 9, 11. Second differences: +2 throughout. Quadratic; T_n = n² + 1.

3. Terms: 1, 1/3. Ratios: 1/3 throughout. Exponential; T_n = 81(1/3)^n-1.

4. Terms: 0.1875, 0.09375, 0.046875. Ratio: 0.5. Exponential.

5. Terms: 21, 25. First difference +4. Linear; possible rule T_n = 4n + 1.

6. Terms: 25, 36. First differences: 3, 5, 7, 9. Second difference: +2. Quadratic; T_n = n².

7. Terms: 80, 160. Ratio 2. Exponential; T_n = 5(2^n-1).

8. Terms: 8, 5. First difference −3. Linear.

9. Terms: 31, 43. First differences: 4, 6, 8, 10, 12. Second difference: +2. Quadratic; a possible rule is T_n = n² + n + 1.

10. One possible answer: the next term could be 10 under a constant-difference rule, but another rule could produce a different term. More terms or a stated rule would help.

11. Missing term: 14. First difference is +4; a possible rule is T_n = 4n + 2.

12. A possible rule is T_n = (n + 1)². It gives 4 when n = 1 and 25 when n = 4.