Sequence Find the general term formula, how to find the general term formula for the number series

Is it (an-1) or (an-1)+1 under the score line?

1. Constructing the method of equal difference number series.

2. Constructing an equal proportional sequence method:

1 Define the construction method. By defining q=a(n+1) an, the proportional sequence is constructed through transformation.

is constant), which can be reduced to a proportional sequence of a(n+1)+x=a(an+x).

is constant). It can be constructed as a proportional sequence of a(n+1)+x*c(n+1)=a(an+x*c n).

into a proportional sequence of a(n+1)+x1n+x2=a(an+x1(n-1)+x2).

3. Function constructor.

For some more complex recursive formulas, by analyzing the structure, associating with formulas and functions similar to the recursive structure, and then constructing a bridge function to find the general term formula of the recursive formula.

For example, the number series an, a1=1, a(n+1)=an 3-3an, find the general term formula an.

Solution Consider the recursive relationship with the formula (a+b) 3=a 3+3a 2b+3ab 2+b 3.

Write out the first 5 items first, find the pattern...

This is not the first big question in the 2017 national college entrance examination number series in Paper 3, which is easy to find. In the second question, you can use the first n terms and sn of the proportional series

In general, the general formula for finding a sequence of numbers has the following principles:

1) If there are positive and negative in the known number series, then determine the positive and negative signs first, and generally use (-1) n or (-1) (n-1) to represent the positive and negative signs.

where (-1) n indicates that the odd term is negative, and the other indicates that the odd term is positive, and 2) after the positive and negative signs are determined, the plus and minus signs are no longer considered, and the general terms are just found for the rest.

If there are both integers and fractions in a given sequence, then the integers must be written as fractions, and then the numerator and denominator can be separated to find the general term.

3) When the given series of numbers are integers, generally see whether the sum or difference between the adjacent two terms is the same, if it is different, there is a certain rule, such as the n power of a certain number, etc., if the above is not good, then look at the difference between the two The general term of the series is found first, and then the accumulation method can be used to find the general term of the original series.

Solution: s2 = 2 2 * a2 = a1 + a2 = 1 2 + a2

a2 = 1/6

s3 = 3^2 * a3 = a1 + a2 + a3 = 1/2 + 1/6 + a3

a3 = 1/12

s4 = 4^2 * a4 = a1 + a2 + a3 + a4 = 1/2 + 1/6 + 1/12 + a4

a4 = 1/20

The general formula for guessing is an = 1 [n(n+1)].

Certificate: When n = 2, there is.

s2 = 2^2 * a2 = a1 + a2 = 1/2 + a2

a2 = 1/6 = 1/[2*(2+1)]

Suppose that when n = n, there is an = 1 [n(n+1)], sn = n 2 * an = n (n+1), then.

When n = n+1, there is.

sn+1 = n+1)^2 * an+1 = a1 + a2 + an + an+1 = n/(n+1) +an+1

an+1 = n/(n+1)]/n+1)^2 - 1] =1/[(n+1)(n+2)]

So when n = n+1, the formula holds.

So, for any n, there is an+1 = 1 [(n+1)(n+2)], and the proposition is true.

Proof is complete.