NAME: _____________________________________________________   ID:___________________


DATE: 19 oct 2026


Lecture: Inductive Proofs

Using the Sipser Theorem 0.25 in the section in Chapter 0 - Proof by Induction, pp 22-25, we need to be sure we can follow the math, so as to understand the proof.

P - principal, amount of loan = $100,000
I - yearly interest rate = 6% or 0.06
t - time in months, starting from t=0
M - monthly multiplier = 1 + I/12 = 1 + .06/12 = 1 + .005 = 1.005
P_t - amount of loan outstanding after t th month
Y - $1000 monthly payment

I. Proof of base case (p24):  use t=0 in the formula for P_t

Get P_0 = P, which holds because we defined P_0 as P.

Agreed?



II. Proof of induction step (p24-25): For each k >= 0, assume the formula P_t is true for t=k.  Show it is true for t = k+1.

1. Write the p25 formula for induction hypothesis:

     P_k = 




Write the proof for t=2 given t=1 has these numbers (including M=1.005):

For t=1
P_1 = P M_1 - Y (M_1 - 1 / M - 1)

P_1 = $100,000 (1.005)  -  $1000 (  )

P_1 = $100,500 - $1000

P_1 = $99,500


For t=2, compute the numbers, assume the IH and show the proof.  Copy the formulas, write in the numbers.