The formula of equal principal and interest repayment is derived, assuming that the total loan amount is A, the annual interest rate of the bank is β, the total number of installments is M (year) and the annual repayment amount is X,
Loans owed to banks at the end of each year are:
The first year-end A( 1+β)-X
At the end of the second year [a (1+β)-x] (1+β)-x = a (1+β) 2-x [1+β].
At the end of the third year {[a (1+β)-x] (1+β)-x} (1+β)-x = a (1+β) 3-x [1+]
…
It can be concluded that the bank loans owed after the nth year are:
a( 1+β)^n-x[ 1+( 1+β)+( 1+β)^2+…+( 1+β)^(n- 1）=a( 1+β)^n-x[( 1+β)^n- 1]/β
Because the total repayment period is m, that is, all bank loans have just been paid off in M, there are:
a( 1+β)^m-x[( 1+β)^m- 1]/β= 0
From this, it can be concluded that:
x = aβ( 1+β)^m/[( 1+β)^m- 1】
Note: 1. In the geometric series of this formula, (1+β) can be regarded as q, and m- 1 is the power of (1+β). However, if we refer to the summation formula sn = a1(1-q n)/(1-q) [which can be quoted here as sn = a1(q n-1)/(q-/kloc-0
2. Monthly repayment also applies to this formula. Note that β is the monthly interest rate at this time, which can be converted by year, and m is the number of repayment months, that is, 1 year equals 12 months.
Answer the question: x = a β (1+β) m/[(1+β) m-1] = 200×10% (1+10%.