1, definition:

2. Sum formula:

3. General formula:

4. From the definition of geometric series, the general term formula and the first n terms formula, we can deduce that:

Arithmetic series formula:

1, definition

For series, if:

Then this series is called arithmetic progression. Where the tolerance d is a constant and n is a positive integer.

2. General formula

An = a1+(n-1) * D. The first item a 1= 1, and the tolerance d=2.

3. The sum of the first n items is sn = a1* n+[n * (n-1) * d]/2.

Sn=[n*(a 1+an)]/2

Sn=d/2*n？ +(a 1-d/2)*n

Extended data:

Geometric series are often used in life. Banks have a way to pay interest-compound interest. That is, the interest of the previous period and the principal are added together as the principal to calculate the interest of the next period, which is what people usually call "rolling interest". The formula for calculating the sum of principal and interest according to compound interest: sum of principal and interest = principal *( 1+ interest rate) deposit period.

With the rise of house prices, many people can't pay off their house prices in one lump sum like this. They always have to borrow money from banks. They can apply for provident fund or bank loans. However, if they want to know how much principal they need to repay after a certain period of time, they can also use this series to calculate for themselves.

As we all know, mortgage loans (provident fund loans) generally implement equal monthly repayment of principal and interest. Let's find a solution to this problem.

If the loan amount is a0 yuan, the monthly interest rate of the loan is P, and the repayment method is equal to the monthly repayment of the principal and interest of A yuan, let the principal after the nth repayment be an.

Then there is: a1= A0 (1+p)-a; a2 = a 1( 1+p)-a； a3 = a2( 1+p)-a； ..... an+ 1=an( 1+p)-a, ..... and transform it to get (an+1-a/p)/(an-a/p) =1+p.

It can be seen that {an-a/p} is a geometric series with a 1-a/p as the first term and 1+p as the common ratio.

In fact, similar bank deposits and loans include lump-sum deposits and withdrawals, which can even be extended to cell division in the biological world.

Baidu encyclopedia-geometric series

Baidu Encyclopedia-arithmetic progression Formula