Can this recursive formula be conferted to an explicit?

Question

This is the recursive formula I derived for the amount of money in an account as the user withdraws $30 each month and gains a 16% annual interest compounded monthly.  I have simplified the compound interest equation as much as I can:

A(o) = 2165.43

A(n) = 76/75 A(n-1) - 30.4

Note that the n and the n-1 are subscripts denoting the term.  I was wondering if my recursive formula could be converted to an explicit formula.  I have not yet found one.  Thanks in advance.

nle · Accepted Answer

yes , you can

Here : A(n) = -114.57*(76/75)^n+2280

try to see if it works

hickerson · Answer

For the 1st series: I observe that the type between the numbers are 3, 5, 7, 9, 11, as a result: f(one million) = 0 f(n) = f(n-one million) + 2n - one million as a result, the series is quadratic, i.e.: f(n) = an^2 + bn + c as a result: 0 = a(one million)^2 + b(one million) + c = a + b + c 3 = a(2)^2 + b(2) + c = 4a + 2b + c 8 = a(3)^2 + b(3) + c = 9a + 3b + c fixing that, i'm getting: a = one million b = 0 c = -one million as a result: f(n) = n^2 - one million For the 1st series: I observe that the type between the numbers is often 3, as a result: f(one million) = one million f(n) = f(n-one million) + 3 as a result: f(n) = mn + b as a result: one million = m(one million) + b 4 = m(2) + b fixing that, i'm getting: m = 3 b = -2 as a result: f(n) = 3n - 2

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Can this recursive formula be conferted to an explicit?

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