Since our recursion uses the two previous terms, our recursive formulas must specify the first two terms. It turns out that each term is the product of the two previous terms. Solution The terms of this sequence are getting large very quickly, which suggests that we may be using either multiplication or exponents. Since our recursion involves two previous terms, we need to specify the value of the first two terms:Įxample 4: Write recursive equations for the sequence 2, 3, 6, 18, 108, 1944, 209952. Each term is the sum of the two previous terms.
Each of these numbers can also be called a term. Solution: This sequence is called the Fibonacci Sequence. A sequence is, simply put, a list of numbers. Solution: The first term is 2, and each term after that is twice the previous term, so the equations are:Įxample 3: Write recursive equations for the sequence 1, 1, 2, 3, 5, 8, 13. Notice that we had to specify n > 1, because if n = 1, there is no previous term!Įxample 2: Write recursive equations for the sequence 2, 4, 8, 16. Solution: The first term of the sequence is 5, and each term is 2 more than the previous term, so our equations are:
#RECURSIVE SEQUENCES CALCULATOR HOW TO#
Recursive equations usually come in pairs: the first equation tells us what the first term is, and the second equation tells us how to get the n th term in relation to the previous term (or terms).Įxample 1: Write recursive equations for the sequence 5, 7, 9, 11. If a sequence is recursive, we can write recursive equations for the sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a specific number. Why? In an arithmetic sequence, each term is obtained by adding a specific number to the previous term. If we go with that definition of a recursive sequence, then both arithmetic sequences and geometric sequences are also recursive. Recursion is the process of starting with an element and performing a specific process to obtain the next term. However, when the value calculation is changed a bit to have, for example, vectorized rates and add a fixed vector of Deposits into the account, the algebra becomes onerous and it feels less expressive to say something like: AccountValue <- InitialValue*cumprod((1+r.vec)(1-w.vec)) + cumsum(Deposits.vec/cumprod((1+r.vec)(1-w.vec)))*cumprod((1+r.vec)(1-w.We've looked at both arithmetic sequences and geometric sequences let's wrap things up by exploring recursive sequences. So in R we would use arrays to flatten it out and simply write: AccountValue <- InitialValue*((1+r)*(1-w))^(1:T) In this example, clearly we could instead say: AccountValue = AccountValue*((1+r)*(1-w))^(i) One R-ish way of handling this is to use algebra to turn the dynamic dependence into a recursive sequence, and then flatten out the sequence to a static formula. The interest rate r and withdrawal rate w are intended to be vectors rather than constants, but I present them as constants to further illustrate my question. So we have the following dynamic formulae: EarnedInterest = r*AccountValueĪccountValue = AccountValue + EarnedInterest - WithdrawnAmountĪccount value is typically seeded with a non-zero initial amount. My specific example is creating an annuity table given a vector of interest rates, withdrawal rates, and fees. I'm wondering if R has a Iterator - like map, lapply, apply, etc - that allows for computing recursive sequences iteratively?
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