IM Alg2.1.8 Lesson: The nᵗʰ Term

Which one doesn’t belong?
A: [math]5+2+2+2+2+2+2[/math][br]B: [math]5+6\cdot2[/math][br]C: [math]5\cdot2^6[/math][br]D: [math]5\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/math]
Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again. . .
Instead of writing a recursive definition, Clare writes [math]C\left(n\right)=80\cdot\left(\frac{1}{2}\right)^n[/math], where [math]C[/math] is the area, in square inches, of the paper after [math]n[/math] cuts. Explain where the different terms in her expression came from.[br]
Approximately what is the area of the paper after 10 cuts?[br]
Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away one inch of the width. Then he does it again, and again. . .
Complete the table for the area of Kiran’s paper [math]K(n)[/math], in square inches, after [math]n[/math] cuts.
Kiran says the area after 6 cuts, in square inches, is [math]80-8\cdot6[/math]. Explain where the different terms in his expression came from.[br]
Write a definition for [math]K\left(n\right)[/math] that is not recursive.[br]
[size=150]Which is larger, [math]K(6)[/math] or [math]C(6)[/math]?[/size]
A Sierpinski triangle can be created by starting with an equilateral triangle, breaking the triangle into 4 congruent equilateral triangles, and then removing the middle triangle. Starting from a single black equilateral triangle:
[img]data:image/png;base64,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[/img][br][br]Let [math]S[/math] be the number of black triangles in Step [math]n[/math]. Define [math]S(n)[/math] recursively.
Andre and Lin are asked to write an equation for [math]S[/math] that isn't recursive. Andre writes [math]S\left(n\right)=3^n[/math] for [math]n\ge0[/math] while Lin writes [math]S\left(n\right)=3^{n-1}[/math] for [math]n\ge1[/math]. Whose equation do you think is correct? Explain or show your reasoning.[br]
Here is a geometric sequence.
3, ____, 6, ____, 12, ____, 24[br][br]Find the missing terms.
Close

Information: IM Alg2.1.8 Lesson: The nᵗʰ Term