A formula for finding a group of number rules in high school mathematics

A formula for finding a group of number rules in high school mathematics

The general formula of arithmetic sequence is as follows:
an=a1+(n-1)d (1)
The first n terms and formula are as follows:
Sn = Na1 + n (n-1) d / 2 or Sn = n (a1 + an) / 2 (2)
Equal ratio sequence
If the ratio of each term to its previous term is equal to the same constant from the second term, the sequence is called equal ratio sequence. This constant is called the common ratio of equal ratio sequence, which is usually represented by the letter Q
(1) The general formula of equal ratio sequence is: an = A1 * q ^ (n-1)
(2) The first N-term sum formula is: SN = [A1 (1-Q ^ n)] / (1-Q)
The relation between any two terms am and an is an = am · qn-m
(3) From the definition of equal ratio sequence, general term formula, first n term and formula, we can deduce: A1 · an = A2 · an-1 = A3 · An-2 = =ak·an-k+1,k∈{1,2,… ,n}
(4) If m, N, P, Q ∈ n *, then AP · AQ = am · an,
The equal proportion median: AQ · AP = 2AR, AR is AP, AQ equal proportion median
Let π n = A1 · A2 An, then π 2N-1 = (an) 2N-1, π 2n + 1 = (an + 1) 2n + 1
In addition, an arithmetic sequence with all positive items takes the same base number to form an arithmetic sequence; conversely, an arithmetic sequence with any positive number C as the base and the items of an arithmetic sequence as the index to construct the power can is an arithmetic sequence. In this sense, we say that a positive arithmetic sequence and an arithmetic sequence are "isomorphic"
nature:
① If m, N, P, Q ∈ N and M + n = P + Q, then am · an = AP * aq;
② In the equal ratio sequence, the sum of every k items in turn is still equal ratio sequence
"G is the equal proportion of a and B", "G ^ 2 = AB (g ≠ 0)"
In the equal ratio sequence, the first term A1 and the common ratio Q are not zero
Note: in the above formula, a ^ n represents the nth power of A