Your friend lends you \$100. He adds a \$0.25 fee every single day, and then charges you 1% interest. Create a formula that represents this relationship.
The first day:
$T_1=(100+0.25)*1.01=\$101.2525$
And the geometric formula I came up with:
$T_d=(a_{d-1}+0.25)*1.01$ for $d>1$
But now the question is asking me to come up with a formula in terms of $P$ principle and $d$ days. And check the amount after 365 days.
I used Excel, but there is no common ratio (probably because it increases by \$0.25 a day). At first, the ratio is 1.012493766, then 1.012462994, etc. It seems the limit is tending towards 1.01 (after extending it in Excel for over 350 days).
So then I came up with the recursive formula:
$T_d=P(1.0125^{d})$
But it doesn't seem to be close to any of the values I have. The first day is \$101.2525, the 2nd day is \$102.517525. Even rounding the dollar values and using the limit ratio of 1.01 doesn't match. I just can't seem to take into account the $0.25 in the recursive formula.
Your geometric formula is correct. Let us further simplify it as: $$ T_d = (T_{d-1}+0.25)*1.01$$ $$=((T_{d-2}+0.25)*1.01+0.25)*1.01=(1.01T_{d-2}+0.25*1.01+0.25)*1.01$$ $$=(1.01)^2T_{d-2}+(0.25)*(1.01)^2+(0.25)*(1.01)$$ $$=(1.01)^2[(T_{d-3}+0.25)*1.01]+0.25[1.01+1.01^2]$$ $$=(1.01)^3T_{d-3}+0.25[1.01+1.01^2+1.01^3]$$ $$=…$$ $$=(1.01)^{d-1}T_1+0.25[1.01+1.01^2+…1.01^{d-1}]$$
Can you take it from here?