Recurrence Relations (Leaving Cert Applied Maths): Flashcards

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Recurrence Relations
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Recurrence relation definition

Shows how each term relates to the previous term

Recurrence relation vs general term: key difference

Depends on previous term, not position number

Common notation for recurrence relations

Often uses 'u' instead of 'T'

Meaning of Tn=3Tn−1T_n = 3T_{n-1}

Term equals 3 times the previous term

Meaning of Tn+1=Tn−4T_{n+1} = T_n - 4

Term equals previous term minus 4

Meaning of un+1=12un−3u_{n+1} = \frac{1}{2}u_n - 3

Term equals half previous term minus 3

First 4 terms: Tn+1=14TnT_{n+1} = \frac{1}{4}T_n, T1=144T_1 = 144

144,36,9,94144, 36, 9, \frac{9}{4}

Common ratio for 144,36,9,94144, 36, 9, \frac{9}{4}

r=14r = \frac{1}{4}

Formula: infinite sum geometric series (r<1r < 1)

S∞=a1−rS_{\infty} = \frac{a}{1-r}

S∞S_{\infty} for 144,36,9,94,...144, 36, 9, \frac{9}{4}, ...

192192

Main limitation of recurrence relations

Must calculate all previous terms to find a specific term

What difference equations allow you to find

General term formula from recurrence relation

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