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While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. For that, a closed or explicit form for the relationship is preferred. An explicit equation allows us to calculate Pn directly, without needing to know Pn-1. While you may already be able to guess the explicit equation, let us derive it from the recursive formula. We can do so by selectively not simplifying as we go. Fill in the blanks. P₁ = 437 +32 P₂ = P₁ +32 = 437 +32 +32 P3 = P2 + 32 = (437 + 2(32)) + 32 P4 = P3+32 = (437 + 3(32)) + 32 Pn = 437 + _(32) = = + 437 + = 437 + 437 + = 437 + = n In general, an explicit linear equation will be in the form: Pn (32) (32) (32) (32) = a) Use the explicit form of the equation to calculate how many bottles he'll have after 5 years. b) Use the explicit form of the equation to calculate how many bottles he'll have after 1000 years. c) Use Desmos to create a graph like the one shown that models the number of bottles Marco will have after n years. Bottles 700 600 500 400 300 200 100 0 0 1 2 3 Years from now 4 5

While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. For that, a closed or explicit form for the relationship is preferred. An explicit equation allows us to calculate Pn directly, without needing to know Pn-1. While you may already be able to guess the explicit equation, let us derive it from the recursive formula. We can do so by selectively not simplifying as we go. Fill in the blanks. P₁ = 437 +32 P₂ = P₁ +32 = 437 +32 +32 P3 = P2 + 32 = (437 + 2(32)) + 32 P4 = P3+32 = (437 + 3(32)) + 32 Pn = 437 + _(32) = = + 437 + = 437 + 437 + = 437 + = n In general, an explicit linear equation will be in the form: Pn (32) (32) (32) (32) = a) Use the explicit form of the equation to calculate how many bottles he'll have after 5 years. b) Use the explicit form of the equation to calculate how many bottles he'll have after 1000 years. c) Use Desmos to create a graph like the one shown that models the number of bottles Marco will have after n years. Bottles 700 600 500 400 300 200 100 0 0 1 2 3 Years from now 4 5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. For that, a closed or explicit form for the relationship is preferred. An explicit equation allows us to calculate Pn directly, without needing to know Pn-1. While you may already be able to guess the explicit equation, let us derive it from the recursive formula. We can do so by selectively not simplifying as we go. Fill in the blanks. P₁ = 437 +32 P₂ = P₁+ 32= 437 +32 +32 P3 = P2 + 32 = (437 + 2(32)) + 32 P4 = P3+32 = (437 + 3(32)) + 32 Pn = 437 + _(32) = = + 437 + = 437 + 437 + = 437 + = n In general, an explicit linear equation will be in the form: Pn (32) (32) (32) (32) = a) Use the explicit form of the equation to calculate how many bottles he'll have after 5 years. b) Use the explicit form of the equation to calculate how many bottles he'll have after 1000 years. c) Use Desmos to create a graph like the one shown that models the number of bottles Marco will have after n years. Bottles 700 600 500 400 300 200 100 0 0 1 2 3 Years from now 4 5
Transcribed Image Text:While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. For that, a closed or explicit form for the relationship is preferred. An explicit equation allows us to calculate Pn directly, without needing to know Pn-1. While you may already be able to guess the explicit equation, let us derive it from the recursive formula. We can do so by selectively not simplifying as we go. Fill in the blanks. P₁ = 437 +32 P₂ = P₁+ 32= 437 +32 +32 P3 = P2 + 32 = (437 + 2(32)) + 32 P4 = P3+32 = (437 + 3(32)) + 32 Pn = 437 + _(32) = = + 437 + = 437 + 437 + = 437 + = n In general, an explicit linear equation will be in the form: Pn (32) (32) (32) (32) = a) Use the explicit form of the equation to calculate how many bottles he'll have after 5 years. b) Use the explicit form of the equation to calculate how many bottles he'll have after 1000 years. c) Use Desmos to create a graph like the one shown that models the number of bottles Marco will have after n years. Bottles 700 600 500 400 300 200 100 0 0 1 2 3 Years from now 4 5
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