Assuming that the total loan amount is A, the monthly interest rate of the bank is β, the total number of installments is M (months) and the monthly repayment amount is X, the monthly loan owed to the bank is:

First month

The second month a (1+β)-x

The third month (a (1+β)-x) (1+β)-x = a (1+β) 2-x [1+β)]

The fourth month ((a (1+β)-x) (1+β)-x) = a (1+β) 3-x [1+(65438).

…

It can be concluded that the bank loan owed after the nth month is

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 repaid in the m th month, so there is

a( 1+β)m–X[( 1+β)m- 1]/β= 0

Thus obtained

x = aβ( 1+β)m/[( 1+β)m- 1]

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On a (1+β) n–x [1+(1+β)+(1+β) 2+…+(1+β) n-1]

1,(1+β), (1+β)2, …), (1+β)n- 1 is geometric progression.

Some properties of geometric series

(1) geometric series: an+ 1/an = q, where n is a natural number.

(2) General formula: an = a1* q (n-1);

Generalization: an = am q (n-m);

(3) summation formula: Sn=nA 1(q= 1)

sn=[a 1( 1-q^n)]/( 1-q)

(4) nature:

(1) if m, n, p, q∈N, m+n = p+q, then am an = AP * AQ；;

(2) In geometric series, every k term is added in turn and still becomes a geometric series.

(5) "G is the proportional average of A and B" and "G 2 = AB (G ≠ 0)".

(6) In geometric series, the first term A 1 and the common ratio q are not zero.

So1+(1+β)+(1+β) 2+…+(1+β) n-1= [(1+β) n-655.

Matching principal repayment is different from matching repayment.

Q: What does equal principal repayment mean? Is equal repayment of principal more economical than equal repayment?

Answer: The calculation formula of average capital repayment method is as follows: monthly repayment amount =P/(n× 12)+ total remaining loan amount× I, where p is the loan principal, I is the monthly interest rate and n is the loan term. We can't simply compare the two repayment methods.

Calculation formula of equal repayment

Monthly repayment amount = (principal × monthly interest rate× (1+monthly interest rate) loan months) ÷[( 1+ monthly interest rate) repayment months-1]

In which: monthly interest = residual principal × monthly interest rate of loan.

Monthly principal = monthly contribution-monthly interest

Calculation principle: from the beginning of monthly contribution, the bank collects the interest of the remaining principal first, and then the principal; Interest is paid monthly.

With the decrease of the remaining principal, the proportion of principal in the monthly payment increases, but the monthly payment

The total amount remains unchanged.

Calculation formula of monthly decreasing repayment amount

Monthly principal and interest repayment amount = (principal/repayment months)+(principal-accumulated repaid principal) × monthly interest rate.

Monthly principal = total principal/repayment months

Monthly interest = (principal-accumulated principal repayment) × monthly interest rate

Calculation principle: the amount of principal returned every month is always the same, and the interest decreases with the decrease of the remaining principal.