anonymous
  • anonymous
How do you write a recursive and explicit equation for Arithmetic sequences?
Algebra
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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mathmate
  • mathmate
A recursive equation has the "unknown" on the right hand side. For example, the sequence is 1,2,3,4,5,6,.... and \(T_1=1\). Then we can say that \(T_n=T_{n-1}\), given that \(T_1=1\) . A recursive equation must provide a boundary condition (for example, \(T_1=1\) , for the sequence to start (or end). A famous example of a recursive sequence is the fibonacci numbers, where \(F_n=F_{n-1}+F_{n-2}\) given \(F_0=0,~and~ F_1=1\) Note that the Fibonacci numbers does NOT form an arithmetic sequence. An explicit equation tells you exactly what the value is without knowing previous terms. Example: sequence 1,2,3,4,5... \(T_n = n\) While the explicit equation for the Fibonacci sequence is \(\large F_n=\frac{(1+\sqrt 5)^n-(1-\sqrt 5)^n}{2^n \sqrt 5}\) This way, you see how recurrent equations could be simpler, but takes more efforts to solve.
mathmate
  • mathmate
Then we can say that \(Tn=T_{n−1}+1\), given that T1=1 .
mathmate
  • mathmate
So if t1=1, t2=t1+1=2, t3=t2+1=3.... Since we are using a previous term to calculate the present term, this is called recursion.

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