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Cassandra is repaying an installment loan of \$3,500 with 20 equal monthly
payments of \$196 each. what is the annual percentage rate of the loan?

The standard formula for installment amount A, for n periods with interest per period i, and present value (principal) P is
A=P((1+i)^n)i/((1+i)^n-1)
where
n=20,
A=196
P=3500
substitute to get
196=3500i(1+i)^20/((1+i)^20-1)

I do not believe there is an explicit formula to solve for i, so we can solve it by trial and error or by iteration.

Rearrange to give
f(i)=196-3500i(1+i)^20/((1+i)^20-1)
and we look for i that makes f(i)=0.
Make a table:
f(0.005)=11.67
f(0.01)=2.04
f(0.015)=-7.86
Interpolating, we try
f(0.011)=0.0878
f(0.0111)=-0.1086
So give a final try of
f(0.011045)=-0.0005...
The interest per period is 1.1045%
Annual interest
=1.1045*12=13.254% approx.

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