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Arithmetic and Geometric Sequences and their Summation. 14.1 Sequences. arithmetic sequence. geometric sequence. geometric sequence. geometric sequence. Find the next two terms of the following sequences : 2, 5, 8, 11,…… 2, 6, 18, 54, …. 2, 4, 8, 16,……. 5, -25, 125, -625, ….

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14 1 sequences

14.1 Sequences

arithmetic sequence

geometric sequence

geometric sequence

geometric sequence

Find the next two terms of the following sequences :

2, 5, 8, 11,……

2, 6, 18, 54, ….

2, 4, 8, 16,…….

5, -25, 125, -625, ….

3, 4, 6, 9, 13, …….

5, 2, -1, -4, …..

0, sin20o, 2sin30o, 3sin40o

arithmetic sequence

consider the following sequence 1 3 5 7 9 111

14.1 Sequences

Consider the following sequence:1, 3, 5, 7, 9, ….., 111

3 is the second term of the sequence, mathematically,

T(2) = 3 or T2 = 3

1 is the first term of the sequence,mathematically, T(1) = 1 or T1 = 1

5 is the third term of the sequence, mathematically, T(3) = 5 or T3 = 5

111 is the nth term of the sequence, mathematically, T(n) = 111 or Tn = 111

consider the sequence 2 4 8 16

14.1 Sequences

Consider the sequence 2, 4, 8, 16, ….

So, the sequence can be represented by the general term

T(n) = 2n or Tn = 2n

The sequence is formed from timing 2 to the previous term.

slide7

14.2 Arithmetic Sequence

Illustrative Examples

arithmetic means

14.2 Arithmetic Sequence

Arithmetic Means

When a, b and c are three consecutive terms of arithmetic sequence, the middle term b is called the arithmetic mean of a and c.

arithmetic means13

14.2 Arithmetic Sequence

Arithmetic Means

Insert two arithmetic means between 11 and 35.

slide14

14.2 Arithmetic Sequence

Insert two arithmetic means between 11 and 35.

geometric means

14.3 Geometric Sequence

Geometric Means

When x, y and z are three consecutive terms of geometric sequence, the middle term y is called the geometric mean of x and z.

geometric means23

14.3 Geometric Sequence

Geometric Means

Insert two geometric means between 16 and -54.

slide24

14.3 Geometric Sequence

Insert two geometric means between 16 and -54.

let s consider a sequence t 1 t 2 t 3 t 4 t n

14.4 Series

The expression T(1) + T(2) + T(3) +….+ T(n) is called a series. We usually denote the sum of the first n term of a series by the notation S(n).

Let’s consider a sequence : T(1), T(2), T(3), T(4), …., T(n)

arithmetic sequence 2 5 8 11 arithmetic series 2 5 8 11

14.5 Arithmetic Series

Arithmetic Sequence : 2, 5, 8, 11, …

Arithmetic Series : 2 + 5 + 8 + 11 + ….

formula of arithmetic series

14.5 Arithmetic Series

Formula of Arithmetic Series

S(n) = a + a + d + a + 2d + a + 3d + …. + a + (n - 1)d

l

formula of arithmetic series29

14.5 Arithmetic Series

Formula of Arithmetic Series

S(n) = l + l - d + l - 2d + l - 3d + …. + a + d+ a

slide30

14.5 Arithmetic Series

S(n) = a + a + d + a + 2d + a + 3d + ………... + a + (n - 1)d

S(n) = l + l – d + l - 2d + l - 3d + ….+ a + d+ a

2S(n) =(a + l)+(a + l)+(a + l)+(a + l)+….. +(a + l)

2S(n) = n(a + l)

geometric sequence 3 9 27 81 geometric series 3 9 27 81

14.6 Geometric Series

Geometric Sequence : 3, 9, 27, 81, …

Geometric Series : 3 + 9 + 27 + 81

formula of geometric series

14.6 Geometric Series

Formula of Geometric Series

S(n) = a + aR + aR2 +aR3+ …. + aRn-1

formula of geometric series35

14.6 Geometric Series

Formula of Geometric Series

R.S(n) = aR + aR2 + aR3 +aR4+ …. + aRn

s n a ar ar 2 ar 3 ar n 1

Subtracting two series

S(n) = a + aR + aR2 +aR3+ …. + aRn-1

S(n) –R.S(n) = a - aRn

R.S(n) = aR + aR2 + aR3 +aR4+ …. + aRn

(1 – R) S(n) = a (1 – Rn)

slide37

14.6 Geometric Series

Timing –1 on both numerator and denominator

consider such a geometric series

14.6 Geometric Series

Sum to Infinity of a Geometric Series

Consider such a Geometric Series

What is the value of common ratio R ?

slide41

14.6 Geometric Series

Sum to Infinity of a Geometric Series

Consider Rn where n tends to the infinity

slide45

(extension module)

Summation Notation

slide46

Consider the symbol

where T( r ) = 3r + 5

= 3(1) + 5 + 3(2) + 5+3(3)

+ 5 + 3(4) +5

= 50

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