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$y(2) = 2y(2-1)-y(2-2)=2y(1)-y(0) = 2 \\cdot 1 - 0 = 2$

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$y(3) = 2y(3-1)-y_(3-2)=2y(2)-y(1) = 2 \\cdot 2 - 1 = 3$

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$y(4) = 2y(4-1)-y(4-2)=2y(3)-y(2) = 2 \\cdot 3 - 2 = 4$

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So, it looks like $y_5=5, y_6=6,...$ and $y_n=n$.

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Recurrence equations - rekursjon

Fill in for  $n$  i  $y_n = 2y_{n-1}-y_{n-2}$

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$y_2 = 2y_{2-1}-y_{2-2}=2y_1-y_0 = 2 \\cdot 1 - 0 = 2$

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$y_3 = 2y_{3-1}-y_{3-2}=2y_2-y_1 = ...$

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A sequence $y_n$ is defined by the recursive formula

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$y_n = 2y_{n-1}-y_{n-2}$

\n

where

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$y(0)=0,\\;\\;y(1)=1$

\n

Find the next 3 terms by using the recursion method.

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$y(2)$ = [[0]] ,   $y(3)$ = [[1]] ,   $y(4)$ = [[2]]

\n

Can you work out a direct formula for   $y(n)$ = [[3]]

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Recurrence equations - rekursjon

"}, "rulesets": {}, "statement": "", "variables": {}, "parts": [{"marks": 0, "variableReplacements": [], "prompt": "

A sequence $y(n)$ is given by the recursive formula

\n

$y(n) - 6y(n-1) +9y(n-2)= 2^n, \\;\\;y_0=1,\\;\\;y_1=2$

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Find a direct formula for $y(n)$:

\n

$y_n$ = [[0]]

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A sequence $y(n)$ is given by the recursive formula

\n

$y(n) - 3y(n-1) +2y(n-2)= 0, \\;\\;n\\geq3,\\;\\;y(1)=0,\\;\\;y(2)=2$

\n

Find a direct formula for $y(n)$:

\n

$y(n)$ = [[0]]

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Recurrence equations - rekursjon

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Recurrence equations - rekursjon

", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "ungrouped_variables": [], "preamble": {"js": "", "css": ""}, "parts": [{"gaps": [{"showCorrectAnswer": true, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "expectedvariablenames": [], "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "jme", "checkvariablenames": false, "vsetrange": [0, 1], "showpreview": true, "marks": "4", "answer": "5*3^n-2*n-3", "checkingtype": "absdiff"}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "prompt": "

A sequence $y(n)$ is defined by the recursive formula

\n

$y(n) - 3y(n-1) = -4n, \\;y(0)=2$

\n

Find a recurrence formula for $y(n)$:

\n

$y(n)$ = [[0]]

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Recurrence equations - rekursjon

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A sequence $y(n)$ is defined by the recursive formula

\n

$y(n) - 3y(n-1) = 0$ and $y(0)=2$.

\n

Find a recurrence formula for $y(n)$:

\n

$y(n)$ = [[0]]

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First- and second order recurrence equations, homogenous and nonhomogenous

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