Grade 7 Math Puzzle 9: Find and verify a two-term recursive pattern

A free grade 7 math puzzle that teaches Find and verify a two-term recursive pattern. Try it first — the hint and the full solution open when you are ready.

Problem 9 · Find and verify a two-term recursive pattern · 4 points

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One number in this chain has been smudged: 2, 5, 12, 29, __, 169, 408. What number was it?

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Look at groups of three numbers. Can you find a relationship that works on both sides of the blank?
Notice that 12 = 2 × 5 + 2 and 29 = 2 × 12 + 5. What would come after 29 if that relationship continues?

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Answer: B — 70

Each number is twice the previous number plus the number before that. So the blank is 2 × 29 + 12 = 70. This checks in both later terms: 2 × 70 + 29 = 169 and 2 × 169 + 70 = 408. The tempting 63 comes from adding 5 instead of 12 to twice 29.

Grade 7 puzzle 9: Find and verify a two-term recursive pattern

Problem 9 · Find and verify a two-term recursive pattern · 4 points