Question

Question 4 (1 point) Mortgage Payment Formula As payments are made to pay off a mortgage, the principal balance remaining to be paid goes down a little after each payment in a non-linear fashion. The principal balance after k months (k payments) is: P = -m ((r + 1)* - 1 - + (r + 1)*P Where: m is the monthly payment (same amount each month). r is the monthly interest rate as a decimal number, e.g., 0.5%/month = P, is the initial balance to be paid. r = 0.005 (a) Show the derivation of this formula for Pusing the techniques of series / sequence analysis combined with explicit use of Mathematical Induction. Simply retyping derivations found on line will not suffice. (b) After proving the validity of the formula for Pk, find the formula for m used in Pk with the assumption that the mortgage has is paid off after n months. Show all work for credit.

Answer 1

a) Given that the initial balance to be paid is Po Thus, the amount to be paid after one month (that is, aftes just one payment has been made) is Polita) -m. This is because the balance will be Poli+) that payment of m because of interest, less made at the end of tiost month. After the Second payment has been made at aneount the end of second month, the outstending will be (1+) (P.(1++)-m) -M = P,(1+ x)-m-m(1+7) %3D After the third payment has been made at the end of third month, the outstending amount will be (1+2)-m-R(1+1)-m-mli+) -m(1+ 3)2 proceding this way, we conjecture that, at the -m-m (Hi) (Po(147)* end of k months, the autstanding amount will be {(&+1)d n(1+7)-mci+*---- ml140) ml4gj*)

Since Icnow that xk_I +xk-I I+x+2?4- OL-I The above (conjectesed) foomola shows that the outstanding amount at the end of k months is Poli+1)°-m-m (1+)-m(1+x)= --- m(i+)*-P(+m( -m(itx)*-! ((+8)*. That is lIty)k-I Px = Po(1+1)h -m(*-) To poove that this is comect, we use induction For k-1 ity) P, = Po(I+r)_m which is precisely Pu we have found abve. Thos the fomula Pix =(1+8)" Po -m ((a-) is true for k =l. Assume that k>) & that the formola is correct foo all numbers <k, Thus. /!!+)*-_ Px-1= (1+)*B-m Po- But then, aur angument above shows that . > (1+3) PR-1-m Pk = (1+Y) (1+2)*-'P -(1+7)m/(!+)^_)

= (1+)" 8--/0-(1)) (1+3)*. (l+)"Po which is ag in aur conjectured formula. Thos, if the formula hold for k-1, then it holds for k. By înduction, we conclude that the fommula holds for all integess Thus LIA7)A, -m Pis = (1+)P b) Assume etactly n months. Thus, after the last payment the autstanding amount is 0, that is Pnz0.This now that the mostage has been paid off in Pmplies o= (14 Y)8o-m(\*) %3D 8(1+)^ Po (1++)21 ode (-(1+8)n