# annuities and interest

The loan of 30.000 euros is depriciated after 10 years, with equal annuities and interest of 11% of annual capitalization. how can the annuity be determined? -this one seems to be the toughest of them all for me.

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Please do not use "homework" as the only tag for a question. – Firefeather Oct 11 '12 at 22:29

You start with 30000, pay out an annuity $A$, so you have $30000-A$, then credit 11% interest, so you have $$(30000-A)(1.11)=(30000)(1.11)-1.11A$$ Then, you pay out $A$ again, and again credit 11%, so you have $$((30000-A)(1.11)-A)(1.11)=30000(1.11)^2-(1.11)^2A-1.11A$$ Now you do it again, getting $$30000(1.11)^3-((1.11)^3+(1.11)^2+1.11)A$$ After 10 of these episodes, everything's gone. So you want $$30000(1.11)^{10}-((1.11)^{10}+(1.11)^9+\cdots+1.11)A=0$$ Now you can solve this for $A$.
EDIT: If you credit interest before paying annuity, the calculations look like $$1.11P-A,\quad1.11^2P-1.11A-A,\quad1.11^3P-1.11^2A-1.11A-A,\dots$$ where $P=30000$, and you wind up with $$1.11^{10}P-(1.11^9+1.11^8+\cdots+1)A=0$$ to solve for $A$.