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then the sequence kT (1)+ m , kT (2)+ m , kT (3)+ m , ... can be obtained. - 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. More about the properties of arithmetic and geometric sequences:. If T (1), T (2), T (3), ... is an arithmetic sequence, then

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slide1

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
  • kT(1)+m, kT(2)+m, kT(3)+m, ...is also an arithmetic sequence.

18.Arithmetic and Geometric Sequences

Let d be the common difference of the arithmetic sequence

then d = T(n+1)-T(n).

T(1), T(2), T(3), ... ,

Each term of the sequence is multiplied by k,

and m is added,

then the sequence kT(1), kT(2), kT(3), ... can be obtained.

then the sequence kT(1)+m, kT(2)+m, kT(3)+m, ... can be obtained.

Let Q(n) be the general term of the new sequence,

then Q(n+1)-Q(n) = [kT(n+1)+m]-[kT(n)+m]

[kT(n)+m]

Remove the square brackets

= kT(n+1)+m-

kT(n)-m

= k[T(n+1)-T(n)]

Factorize

= kd

∵ kdis a constant

∴kT(1)+m, kT(2)+m, kT(3)+m, ...is also an arithmetic sequence

slide2

More about the properties of arithmetic and geometric sequences:

  • 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.g.

18.Arithmetic and Geometric Sequences

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 itscommon difference.

The common difference of the arithmetic sequence a1, a2, a3, ... is d,

then d = an+1-an.

Each term is multiplied by 3, and 5 is added,

then the sequence 3a1+5, 3a2+5, 3a3+5, ... can be obtained.

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)

= 3an +1+5-3an-5

= 3(an +1-an)

= 3d

slide3

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.

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).

kT(n+1)

kT(n+1)

=

=

T(n+1)

kT(n)

kT(n)

T(n+1)

then r = .

=

T(n)

T(n)

Q(n+1)

then

Q(n)

18.Arithmetic and Geometric Sequences

Let r be the common ratio of the geometric sequence T(1), T(2), T(3), ... ,

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,

Reduce the fraction

The common ratio of the geometric sequence

= r

∴kT(1), kT(2), kT(3), ...is also a geometric sequence

slide4

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).

E.g.

64

=

16

∴T(n) =

4(4)n-1 = 4n

18.Arithmetic and Geometric Sequences

(i) Prove that 4, 16, 64, 256, 1 024, ... is a geometric sequence. Hence, find the general term T(n) of the sequence.

T(3)

T(4)

T(2)

T(3)

T(2)

T(4)

256

16

Prove:

=

=

=

=

=

=

4

4

4

4

=

=

T(3)

T(1)

T(2)

T(1)

64

4

T(2)

T(3)

The common ratio of the geometric sequence

∴ 4, 16, 64, 256, 1 024, ...is a geometric sequence

slide5

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).

E.g.

18.Arithmetic and Geometric Sequences

(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(2)

Q(5)

Q(3)

Q(4)

4

=

=

=

=

Q(1)

Q(4)

Q(2)

Q(3)

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,

8, 32, 128, 512, 2 048, ... is also a geometric sequence.

The common ratio of the geometric sequence

The common ratio of the new sequence is 4.

∴Q(n) = 8(4)n-1= 2(4)n