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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} …
s1=a
s2=2a+d
s3=3a+3d
s2^2=s1*s3
(2a+d)^2=a*(3a+3d)
4a^2+4ad+d^2=3a^2+3ad
a^2+ad+d^2=0
Then solve the equation to get D
If the sum of the first n terms of the sequence an is Sn = -- N ^ 2, find the general term formula of the sequence, and judge whether the sequence is equal difference sequence or equal ratio sequence, please give the proof
Sn=-n^2
a1=S1=-1
When n ≥ 2
an=Sn-S(n-1)=-n^2-[-(n-1)^2]=-n^2+n^2-2n+1=1-2n
Where a1 = - 1 = 1-2 * 1 also conforms to the general formula
So an = 1-2n is an arithmetic sequence
The sum of the first n terms of the sequence (an} is Sn, and satisfies an = - 3Sn * sn-1 (n > = 2), A1 = 1 / 3 1. It is proved that (1 / Sn} is an arithmetic sequence
The sum of the first n terms of the sequence (an} is Sn, and satisfies an = - 3Sn * sn-1 (n > = 2), A1 = 1 / 3
1. Prove that (1 / Sn} is arithmetic sequence
2. Let BN = 2 (1-N) an (n > = 2) prove B2 square + B3 square +. + BN square
1）S[n]-S[n-1] = -3S[n]*S[n-1]
Divide two sides by - s [n] * s [n-1]
1/S[n] -1/S[n-1] =3
2）S[n]=1/(3n)
a[n]= S[n] - S[n-1] =1/( 3(1-n)n )
b[n]=2/(3n)
Because 1 / (n ^ 2) < 1 / ((n-1) n) = 1 / (n-1) - 1 / n
B2 square + B3 square +. + BN square
=4/9*(1/2^2 +1/3^2+...+1/n^2)
It is known that the sum of the first n terms of the sequence {an} is Sn, and A1 = 2, 3Sn = 5an-an-1 + 3sn-1 (n ≥ 2, n ∈ n *) (I) find the general formula of the sequence {an}; (II) let BN = (2n-1) an, find the sum of the first n terms of the sequence {BN}
(I) by 3Sn = 5an-an-1 + 3sn-1 3an = 5an-an-1 (n ≥ 2, n ∈ n *), Anan − 1 = 12, (n ≥ 2, n ∈ n *), so the sequence {an} is an equal ratio sequence with 2 as the first term and 12 as the common ratio, ｜ an = 22-n (Ⅱ) BN = (2n-1) · 22-n TN = 1 × 2 + 3 × 20 + 5 × 2-1 + + (2n-1) · 22
RELATED INFORMATIONS
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- 3. 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
- 4. Given the arithmetic sequence {an}, A1 = 1, an = a (n + 1) + 2 (n ≥ 1), then what is A100?
- 5. In the arithmetic sequence an, a1 + A2 +. + A50 = 200, A51 + a52 +... + A100 = 2700, then the value of A1 is?
- 6. In the arithmetic sequence {an}, a1 + A2 + +a50=200，a51+a52+… +If A100 = 2700, then A1 equals () A. -1221B. -21.5C. -20.5D. -20
- 7. As shown in the figure, in the rectangular coordinate system, the coordinates of point P are (3,4). Rotate OP 90 ° counterclockwise around the origin o to get the line segment op ′. (1) draw the line segment op ′; (2) find the coordinates of P ′ and the length of PP ′
- 8. In the plane rectangular coordinate system, the coordinate of the point (1,2) obtained by translating the point (1,2) to the right unit length is
- 9. In the plane rectangular coordinate system, the point a (x, 2) is translated up three unit lengths, and then to the right two unit lengths to get the point B (- 3, y), then x =, y=
- 10. In △ ABC, ∠ C = 90 °, B = 30 ° and BC = 3, then () A. B. C. D.
- 11. 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
- 12. In the known sequence (an), a1 + A2 = 9, a1a2a3 = 27, Sn =? I still can't find A1 and Q. how can I find them
- 13. In {an}, an + 1 > an, a1 + A4 = 9, A2 * A3 = 8, S10 =?
- 14. In the arithmetic sequence {an}, if A1, A3 and A4 form an arithmetic sequence, then the common ratio of the arithmetic sequence is______ ．
- 15. 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
- 16. 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
- 17. 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
- 18. 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
- 19. In the arithmetic sequence {an}, a1 + a3 + A5 = 18, an-4 + An-2 + an = 108, Sn = 420, then n=______ ．
- 20. 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