PLEASE HELP ??? :))

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## 1 Answer

The easiest way to do this is to use an iterative process.

P[n]=P[n-1](1+r)-m

where P is the current balance on the loan so that P[0]=\$1100, n=month number starting at zero, m=monthly payment, r=monthly rate.

So m=\$71.50, r=19.2/12%=1.6%=0.016.

Therefore P[n]=1.016P[n-1]-71.50.

The table below shows the decreasing balance:

 n Balance (\$) 0 1100.00 1 1046.10 2 991.34 3 935.70 4 879.17 5 821.74 6 763.38 7 704.10 8 643.86 9 582.67 10 520.49 11 457.32 12 393.13 13 327.92 14 261.67 15 194.36 16 125.97 17 56.48

At month 17 the balance is less than the monthly payment, so there are 18 payments in all, but the last payment is reduced to \$56.48.

Algebraically the balance is P[n]=P(1+r)ⁿ-m((1+r)ⁿ-1)/r where P= initial loan.

From this, n=log(m/(m-rP))/log(1+r). Plugging in the values we get n=17.8 months. The table seems to confirm this if P[n]=0. If R=1+r, PRⁿ=m(Rⁿ-1)/r; rPRⁿ/m=Rⁿ-1; Rⁿ(1-rP/m)=1; Rⁿ=m/(m-rP).

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