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In the sequence {an}, A1 = 3, an = 2A (n-1) + n-2 (n is equal to 2, and N belongs to n positive), it is proved that the sequence {an + n} is an equal ratio sequence In the sequence {an}, A1 = 3, an = 2A (n-1) + n-2 (n is greater than or equal to 2, and N belongs to n positive), it is proved that the sequence {an + n} is an equal ratio sequence, and the general term formula of {an} is obtained
Add n on both sides to get a (n) - n = 2 [a (n-1) + n-1] (this is a recursive formula), so {an + n} is an equal ratio sequence, so a (n) + n = 2 ^ (n-1) (a1 + 1) = 2 ^ (n + 1), that is, a (n) = 2 ^ (n + 1) - n ^
In the sequence, A1 = 1, an = 2S (n-1) + 1 (n is greater than or equal to 2) is proved to be equal ratio sequence, and the common ratio is obtained
an=2S(n-1)+1--(1)
a(n+1)=2Sn+1--(2)
(1) - (2), got it
a(n+1)-an=2Sn-2S(n-1)=2an
We obtain a (n + 1) = 3an
So {an} is an equal ratio sequence, and the common ratio is 3
an=3^(n-1)
If the product of the first n terms of an is TN and T5 ^ 6 / T3 ^ 5 = 32, then A4=
The product of the first n terms of the sequence an is TN
T5=a1a2a3a4a5=a3^5
T3=a1a2a3=a2^3
T5^6/T3^5
=a3^30/a2^15
=a3^15*(a3/a2)^15
=a3^15*q^15
=(a3q)^15=32
a4^15=2^5
a4^3=2
A4 = 2 under the triple root
In the equal ratio sequence {an}, the product of the first n terms is TN. if T5 = 1, then the term equal to 1 in {an} is TN
Let the general term an be the N-1 power of A1 times Q, and the first five terms are multiplied to get T5 = the 5th power of A1 times the 10th power of Q, which is the 5th power of A1 times Q square, that is, the 5th power of the whole is equal to 1, that is, A1 times Q square is equal to 1, that is, the third term of the equal ratio sequence
RELATED INFORMATIONS
- 1. In the equal ratio sequence {an}, TN is the product of the first n terms. If T5 = 1, then there must be () A. a1=1B. a3=1C. a4=1D. a5=1
- 2. In the arithmetic sequence {an}, if A1, A3 and A4 form an arithmetic sequence, then the common ratio of the arithmetic sequence is______ ．
- 3. In {an}, an + 1 > an, a1 + A4 = 9, A2 * A3 = 8, S10 =?
- 4. In the known sequence (an), a1 + A2 = 9, a1a2a3 = 27, Sn =? I still can't find A1 and Q. how can I find them
- 5. It is known that SN is the sum of the first n terms of a sequence, and Sn = PN determines whether an is an equal ratio sequence
- 6. Let the first term A1 of the arithmetic sequence {an} be a, and the sum of the first n terms be SN. If s1s2s3 is a proportional sequence, find the general term formula of the sequence {an} …
- 7. The sum of the first n terms of sequence an is Sn A1 = 1 for any n > = 2 3sn-4 an 2-3s (n-1) / 2 General term formula for calculating the value of A2 A3 by assembly arithmetic sequence
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- 9. 1. In the arithmetic sequence {an}, if 2A1 = - A2 is known, then the common ratio is? 2. In the arithmetic sequence {an}, A1 = 3, A100 = 36, then A3 + A98 =? 3. In the sequence {an}, if an + 1-an = 1 / 2 (n = 1,2, ···), and A1 = 2, then A101 =? 4. In the arithmetic sequence {an}, a1 + A2 = 3, A3 + A4 = 6, then A7 + A8 =? 5. If the third and fourth terms of an equal ratio sequence {an} are 4 and 8 respectively, then the first and fifth terms are? 6. Given the sequence {an}, an = PN + Q (P, q are constants, you ∈ n *), where a1 = 2, A17 = 66, find (1) the general term formula an. (2) the first 10 terms and S10
- 10. Given the arithmetic sequence {an}, A1 = 1, an = a (n + 1) + 2 (n ≥ 1), then what is A100?
- 11. The sum of the first n terms of the proportional sequence {an} is Sn if s2n = 3 (a1 + a3 +...) +A2n-1), a1a2a3 = 8, then A10 equals () A. -512B. 1024C. -1024D. 512
- 12. Given that the sum of the first n terms of the equal ratio sequence {an} is Sn = (2 ^ n) - 1, find the sum of the first n terms of the sequence {(an) ^ 2} and TN
- 13. In the arithmetic sequence {an}, a1 + a3 + A5 = 18, an-4 + An-2 + an = 108, Sn = 420, then n=______ ．
- 14. It is known that in the arithmetic sequence an, A4 = 4, the first 10 terms and S10 = 10, BN = (1 / C) are to the power of an, (C is a positive constant.) (1) Find an (2) It is proved that BN is an equal ratio sequence (3) Finding the first n terms and TN of sequence BN
- 15. Let f (x) satisfy 2F (x) - f (1 / x) = 4x-2 / x + 1, sequence {an} and {BN} satisfy A1 = 1, a (n + 1) - 2An = f (n), BN = a (n + 1) - an Finding the analytic expression of F (x) The general term formula of BN Is to compare the size of 2An and BN, and prove that
- 16. Given that the sum of the first n terms of sequence {an} A1 = 2 is Sn and satisfies Sn sn-1 = 3an, the general term formula an of sequence {an} is obtained Given that the sum of the first n terms of the sequence {an} A1 = 2 is Sn and satisfies Sn + sn-1 = 3an, the general term formula an of the sequence {an} is obtained
- 17. In the sequence an, A1 = in, a (n + 1) = 2An + 3n-4, in is a real number. For any in, it is proved that an is not an equal ratio sequence BN = a (n + 1) - an + 3, try to judge whether BN is an equal ratio sequence The formula of finding the same term of an
- 18. It is known that an = 2A (n-1) + 2 ^ n-1 (n ≥ 2) A1 = 5, A2 = 13, A3 = 33, A4 = 81. {(an + α) / 2 ^ n} is an arithmetic sequence, and a is obtained
- 19. It is known that the sum of the first n terms of the sequence {an} is Sn, and a1 + 2A2 + 3a3
- 20. The known sequence {an} satisfies A1 = 1, an = a1 + 2A2 + 3a3 + +(n-1) an-1, then when n ≥ 2, the general term an = () A. n!2B. (n+1)!2C. n!D. （n+1）!