Introduction to Linear Polynomials
Stage n: the general term
Find a rule for the nth stage of a growing pattern.
Fig. 2.4 grows by 2 tiles a stage: 1, 3, 5, 7, so stages 5, 6, 7 have 9, 11, 13 tiles.
"How many at Stage 100, without drawing 100 pictures?" Look for the rule. Each stage has one less than twice the stage number: Stage 2 has 2 × 2 – 1 = 3; Stage 5 has 2 × 5 – 1 = 9. So Stage n has 2n – 1 tiles. It is linear, and the step, 2, is the number in front of n.
Run it backwards: 2n – 1 = 47 means 2n = 48, so n = 24.
Bela has ₹100 and spends ₹5 a day. After n days she has 100 – 5n: ₹40 when n = 12, and ₹0 when n = 20.
An auto fare is ₹25 for the first 2 km, then ₹15 a km. For n km (n ≥ 2): 25 + 15(n – 2) = 15n – 5. So 10 km costs ₹145, and ₹130 buys 9 km.
In every case, the nth term is a linear expression in n. A linear pattern is a sequence where the difference between neighbouring terms is constant.
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