Answer Key and Hints

A. Multiple Choice Questions

1.     (c) 15 — the rule adds one more than the previous addition (1, +2, +3, +4, +5).

2.     (c) 36 — since 36 = 6 × 6.

3.     (b) 64 — the terms are 1, 2, 4, 8, 16, 32, 64 for positions 1 to 7, so the 7th term is 64.

4.     (b) Add the two previous terms.

5.     (b) It lets us find any term directly, without finding every earlier term.

6.     (c) 64 — since 4 × 4 × 4 = 64.

7.     (c) The third diagonal — where the numbers 1, 3, 6, 10 appear in order.

B. Fill in the Blanks

1. rule   2. square   3. row (or: number)   4. Fibonacci   5. Verify   6. odd   7. golden

C. Short Answer Questions — Hints

1.     Any correctly reasoned daily-life example following a rule (e.g., a bus arriving every 15 minutes) is acceptable.

2.     A term-to-term rule connects consecutive terms (e.g., “double it”); a position-to-term rule connects a position number directly to its value (e.g., n × n for squares).

3.     1, 4, 9, 16, 25 — each term is the position number multiplied by itself.

4.     Because the amount of rice doubles on every square, and doubling grows extremely quickly once it has multiplied several times, so early squares look small but later squares hold an enormous number of grains.

5.     Observe, Identify, Predict, Verify.

6.     A gnomon is the L-shaped layer of dots added around a square to form the next larger square; since these gnomons are exactly the odd numbers (1, 3, 5, 7…), every square number is a sum of consecutive odd numbers.

7.     Accept any two of: sunflower seed spirals, pinecone spirals, nautilus shell curves, flower petal counts, or the proportions of certain rectangles and buildings.

D. Short Answer Questions — Hints

1.     The rule is: position n gives n² + 1 (check: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26). For the 8th term: 8² + 1 = 65.

2.     Students should show a triangular arrangement of n dots combined with a triangular arrangement of (n−1) dots, and demonstrate that together they form a complete n × n square.

3.     Answers will vary; accept any well-reasoned example (e.g., a doctor observing symptoms, predicting a diagnosis, and verifying it with a test) showing the four steps applied outside mathematics.

4.     Rows should read 1; 1,1; 1,2,1; 1,3,3,1; 1,4,6,4,1; 1,5,10,10,5,1 — the triangular numbers 1, 3, 6, 10 appear along the third diagonal.

E. Long Answer Questions — Hints

1.     (a) This is the 10th triangular number: 1+2+…+10 = 55 chairs. (b) Find the largest triangular number not exceeding 200: the 19th triangular number is 190 and the 20th is 210, so 19 complete rows can be formed.

2.     Answers should describe multiplicative growth (doubling) versus additive growth from two prior terms (Virahanka), include one real-life example for each (e.g., spread of information for doubling; petal counts or shell spirals for Virahanka), and connect both to Observe-Identify-Predict-Verify as a shared thinking process used in computational thinking and in how AI systems learn from data.

3.     Answers will vary; check that the student's own shape pattern has a consistent, describable rule and that both a term-to-term and a position-to-term rule are correctly stated for at least one of the two shapes used.