Given the following recursive definition is it possible to write it as a sum?

$\displaystyle \displaystyle a_0=1,337,101.43 \ \ \ a_n=a_{n-1}(1+33*.02)*1.03^{n-1}$

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- Mar 26th 2011, 02:35 PMdwsmithRecursive to Sum if possible
Given the following recursive definition is it possible to write it as a sum?

$\displaystyle \displaystyle a_0=1,337,101.43 \ \ \ a_n=a_{n-1}(1+33*.02)*1.03^{n-1}$ - Mar 26th 2011, 03:59 PMHallsofIvy
So $\displaystyle x_n= (1.66)(1.03)^{n-1}a_{n-1}$?

My suggestion would be to calculate some of them:

$\displaystyle a_1= (1.6)a_0$

$\displaystyle a_2= (1.6)(1.03)a_1= (1.6)^2(1.03)a_0$

$\displaystyle a_3= (1.6)(1.03)^2a_2= (1.6)^3(1.03)^3a_0$

$\displaystyle a_4= (1.6)(1.03)^3a_3= (1.6)^4(1.03)^6a_0$

$\displaystyle a_5= (1.6)(1.03)^4a_4= (1.6)^5(1.03)10a_0$

etc.

So it's clear that the formula involves $\displaystyle (1.6)^{n-1}$. The powers on 1.03 are 0, 1, 1+ 2= 3, 1+ 2+ 3= 6, 1+ 2+ 3+ 4= 10, etc. for n= 1, 2, 3, 4, 5, ... That is, of course, $\displaystyle \frac{n(n- 1)}{2}$.