Math of Finance Help Needed??

AdriAnne and Anna鈥檚 Auto Repair wants to add a new service bay. How much can they borrow at an annual rate of 7%, compounded quarterly, if the desired quarterly payment is $6100? Assume a loan period of 4.5 years.

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Let v = 1/(1+i) where i is the interest rate per period. Here i=.07/4.

The formula for the PV of the payments (per $) is
(1-v^n)/i. Here n = 4.5 x 4 = 18. The annuity factor works out to 15.32686. Times 6100 = 93493.86 which is the PV of the payments, or the amount of the loan.

You can check it by adding up v^n for n = 1 to 18. Here's what I had from Excel:

1+i 1.0175
v 0.982800983
i 0.0175
n 18
1-v^n 0.268220098
(1-v^n)/i 15.32686272

1 0.982800983
2 0.965897772
3 0.949285279
4 0.932958506
5 0.916912536
6 0.901142542
7 0.885643776
8 0.870411573
9 0.855441349
10 0.840728599
11 0.826268893
12 0.81205788
13 0.798091283
14 0.784364897
15 0.770874592
16 0.757616307
17 0.744586051
18 0.731779902
sum 15.32686272
6100 93493.86259

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