Given that the sequence {an} satisfies an + 1 = an + 2n + 1, the general term formula of the sequence {an} can be obtained by cumulative addition Note: N and N + 1 next to a are subscripts

Given that the sequence {an} satisfies an + 1 = an + 2n + 1, the general term formula of the sequence {an} can be obtained by cumulative addition Note: N and N + 1 next to a are subscripts

An+1=An+2n+1
A2=A1+2+1
A3=A2+4+1
...
An=(An-1)+2(n-1)+1
A2+..+An=A1+.(An-1)+2+4+..+2(n-1)+1×(n-1)
An=(A1)+2+4+..+2(n-1)+(n-1)=(A1)+n×(n-1)+(n-1)=A1+(n-1)²
An=A1+(n-1)²

Sequence an + an + 1-1 = n (an + 1-an-1), find the general term formula of an. Use the successive difference method

A_ {n}+A_ {n+1}-1=n*(A_ {n+1}-A_ {n-1})-------------------------1
A_ {n-1}+A_ {n}-1=(n-1)*(A_ {n}-A_ {n-2})-------------------------2
Using 1-2, the results are as follows
A_ {n+1}-A_ {n-1}=n*(A_ {n+1}-A_ {n-1})-(n-1)*(A_ {n}-A_ {n-2});
(n-1)*(A_ {n+1}-A_ {n-1})=(n-1)*(A_ {n}-A_ {n-2});
1) N = 1, substituting into Formula 1,
A_ {1}=1;
2) N / = 1
A_ {n+1}-A_ {n-1}=A_ {n}-A_ {n-2}=d;
The tolerance is d
N = 2, with a_ {2}=d.
So the general term is:
A_ {2n}=n*A_ {2};
A_ {2n+1}=n*A_ {2} + 1; (n is a natural number)

Finding the general term formula of sequence 1,3,7,13,21

The general formula is n * (n-1) + 1