Thank you for your answer^~ 1. Given the tolerance D ≠ 0 of the arithmetic sequence {an}, and A1, A3 and A9 are equal proportion sequence, then (a1 + a3 + A9) / (A2 + A4 + A10) =? 2. In the sequence {an}, an + 1 (n + 1 is the angle sign) = an ^ 2 / (2an-5). If the sequence is both equal difference sequence and equal ratio sequence, the general formula of the sequence is______ 3. Given that f (x) = 3x / (x + 3), the sequence {an} satisfies an = f (an-1) (n-1 is a diagonal) (n ≥ 2, n ∈ n *, an ≠ 0), is the sequence {1 / an} equal difference? If so, please prove it and find out its tolerance; if not, please explain the reason 4. Given the distance from the image vertex of the quadratic function f (x) x ^ 2-2 (10-3n) x + 9N ^ 2-61n + 100 (n ∈ n *) to the y-axis to form the sequence {an}, find the general formula of (1) sequence {an} (2) The first n terms and s of sequence {an}

Thank you for your answer^~ 1. Given the tolerance D ≠ 0 of the arithmetic sequence {an}, and A1, A3 and A9 are equal proportion sequence, then (a1 + a3 + A9) / (A2 + A4 + A10) =? 2. In the sequence {an}, an + 1 (n + 1 is the angle sign) = an ^ 2 / (2an-5). If the sequence is both equal difference sequence and equal ratio sequence, the general formula of the sequence is______ 3. Given that f (x) = 3x / (x + 3), the sequence {an} satisfies an = f (an-1) (n-1 is a diagonal) (n ≥ 2, n ∈ n *, an ≠ 0), is the sequence {1 / an} equal difference? If so, please prove it and find out its tolerance; if not, please explain the reason 4. Given the distance from the image vertex of the quadratic function f (x) x ^ 2-2 (10-3n) x + 9N ^ 2-61n + 100 (n ∈ n *) to the y-axis to form the sequence {an}, find the general formula of (1) sequence {an} (2) The first n terms and s of sequence {an}

Do the first question first
① According to the question, A3 ^ 2 = A1 * A9
If it is an arithmetic sequence, then (a1 + 2D) ^ 2 = A1 (a1 + 8D)
The reduction result is A1 = D
（a1+a3+a9）/（a2+a4+a10）=(3a1+10d)/(3a1+13d)=13/16

It is known that the sum of the first four terms of an arithmetic sequence {an} with finite number of terms is 21, and the sum of the last four terms is 67. The sum of all terms of this sequence is 286. How many terms are there in this sequence?
Please write down the reasons

21+67/4=286/n
n=13

High school mathematics! On the sequence of! Online and so on!
Known sequence (an) satisfies A1 = 2, (1 in A1 is the footmark), an + 1 = 0.5an + 3, find an in detail. Thank you
How do you know to subtract six? How did you get 6?

From an + 1 = 0.5an + 3
5 * (an - 6)
Then the sequence (an - 6) is equal ratio sequence
Then when n = 1, a1-6 = 2-6 = - 4
When n 〉 = 2, an - 6 = (a1-6) * (0.5) ^ (n-1)
That is, an = - (1 / 2) ^ (n-3) + 6

1. Given that the sequence satisfies A1 = 1, a (n-1) + 2An = 2, find the general term formula of an
2. It is known that SN is the sum of the first n terms of the sequence {an}, and Sn = 2-2an (n belongs to n *) (1). Proof: the sequence {an} is an equal ratio sequence. (2) find the general formula of the sequence {an}. (3) find the sum of the first n terms of the sequence {ANSN}

1.
An=1-0.5A(n-1)
An-2/3=1/3-0.5A(n-1)=-0.5[A(n-1)-2/3]
The common ratio q = - 0.5, the first term A1-2 / 3 = 1 / 3
An-2/3＝(1/3)*(-0.5)^(n-1)
An＝(1/3)*(-0.5)^(n-1)+2/3
two
When Sn = 2-2an, n = 1, A1 = 2 / 3
S(n-1)＝2-2A(n-1)
That is: an = SN-S (n-1) = 2A (n-1) - 2An
An/A(n-1)＝2/3
The sequence {an} is an equal ratio sequence, and the common ratio is 2 / 3,
An＝(2/3)^n
Sn＝2-2(2/3)^n
ANSN = 2 (2 / 3) ^ n - 2 (4 / 9) ^ n is the difference between two equal ratio sequences!
A1S1＝4/3 - 8/9
Sum of the first n terms = 4 / 3 * [1 - (2 / 3) ^ n] / (1-2 / 3) - 8 / 9 * [1 - (4 / 9) ^ n] / (1-4 / 9)
＝4/9-(4/3)*(2/3)^n+(8/9)*(4/9)^n
For the sake of typing so many words,