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Use the formula of general term to find the common difference of the new sequence. Then, show that the new sequence is also an arithmetic sequence. More about the properties of arithmetic and geometric sequences:. If T (1), T (2), T (3), ... is an arithmetic sequence, then - PowerPoint PPT Presentation
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Use the formula of general term to find the common difference of the new sequence. Then, show that the new sequence is also an arithmetic sequence. then the sequence kT(1)+m, kT(2)+m, kT(3)+m, ... can be obtained.
18.18. Arithmetic and Geometric SequencesArithmetic and Geometric Sequences
then the sequence kT(1), kT(2), kT(3), ... can be obtained.
Let Q(n) be the general term of the new sequence,
∵ kd is a constant
∴ kT(1)+m, kT(2)+m, kT(3)+m, ... is also an arithmetic sequence
Remove the square brackets
Factorize
More about the properties of arithmetic and geometric sequences:(a) If T(1), T(2), T(3), ... is an arithmetic sequence, then kT(1)+m, kT(2)+m, kT(3)+m, ... is also an arithmetic sequence.
= kT(n+1)+m－then Q(n+1) － Q(n) = [kT(n+1)+m] － [kT(n)+m]
= k[T(n+1) － T(n)]
= kd
[kT(n)+m]
kT(n) － m
and m is added,
then d = T(n+1) － T(n).
Let d be the common difference of the arithmetic sequence
T(1), T(2), T(3), ... ,
Each term of the sequence is multiplied by k,
If the common difference of the arithmetic sequence a1, a2, a3, ... is d, prove that 3a1+5, 3a2+5, 3a3+5, ... is an arithmetic sequence, and hence, find its common difference.The common difference of the arithmetic sequence a1, a2, a3, ... is d,
then the sequence 3a1+5, 3a2+5, 3a3+5, ... can be obtained. Each term is multiplied by 3, and 5 is added, then d = an+1 － an.
According to the property of arithmetic sequence,3a1+5, 3a2+5, 3a3+5, ... is also an arithmetic sequence.The common difference of the new sequence
= 3an +1+5 － 3an － 5= 3(an +1 － an)= 3d
18.18. Arithmetic and Geometric SequencesArithmetic and Geometric SequencesMore about the properties of arithmetic and geometric sequences:(a) If T(1), T(2), T(3), ... is an arithmetic sequence, then kT(1)+m, kT(2)+m, kT(3)+m, ... is also an arithmetic sequence.
E.gE.g..
= (3an+1+5) － (3an+5)
Use the formula of general term to find the common ratio of the new sequence. Then, show that the new sequence is also a geometric sequence.Each term of the sequence is multiplied by k (k 0),
then the sequence kT(1), kT(2), kT(3), ... can be obtained.
Let Q(n) be the general term of the new sequence,
∴ kT(1), kT(2), kT(3), ... is also a geometric sequence
Reduce the fraction
The common ratio of the geometric sequence
= r
then Q(n)
Q(n+1) =
kT(n) kT(n+1)
= kT(n)
kT(n+1)
= T(n)
T(n+1)
18.18. Arithmetic and Geometric SequencesArithmetic and Geometric SequencesMore about the properties of arithmetic and geometric sequences:(b) If T(1), T(2), T(3), ... is a geometric sequence, then kT(1), kT(2), kT(3), ... is also a geometric sequence (where k 0).
Let r be the common ratio of the geometric sequence T(1), T(2), T(3), ... , then r = .
T(n) T(n+1)
(i) Prove that 4, 16, 64, 256, 1 024, ... is a geometric sequence. Hence, find the general term T(n) of the sequence.
∴ 4, 16, 64, 256, 1 024, ... is a geometric sequence
Prove:
T(1) T(2)
= 4 16 =
4
T(2) T(3)
=
16 64
=
4 T(3) T(4)
= 64 256
=
4
T(1) T(2)
T(2) T(3)
= T(3) T(4)
= =
4
∴ T(n) = 4(4)n － 1 = 4n
The common ratio of the geometric sequence
18.18. Arithmetic and Geometric SequencesArithmetic and Geometric Sequences
E.gE.g..
More about the properties of arithmetic and geometric sequences:(b) If T(1), T(2), T(3), ... is a geometric sequence, then kT(1), kT(2), kT(3), ... is also a geometric sequence (where k 0).
(ii) If each term of the sequence in part (i) is multiplied by 2, then a new sequence is obtained. Find the general term Q(n) of the new sequence.
∴ Q(n) = 8(4)n － 1 = 2(4)n
Each term is multiplied by 2, then the sequence 8, 32, 128, 512, 2 048, ... can be obtained.
According to the property of geometric sequence,
Q(1) Q(2)
Q(2) Q(3)
= Q(3) Q(4)
= Q(4) Q(5)
= 4 =The common ratio of the geometric sequence
The common ratio of the new sequence is 4.
8, 32, 128, 512, 2 048, ... is also a geometric sequence.
18.18. Arithmetic and Geometric SequencesArithmetic and Geometric Sequences
E.gE.g..
More about the properties of arithmetic and geometric sequences:(b) If T(1), T(2), T(3), ... is a geometric sequence, then kT(1), kT(2), kT(3), ... is also a geometric sequence (where k 0).
