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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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Arithmetic and Geometric Sequences and their Summation

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Arithmetic and geometric sequences and their summation l.jpg

Arithmetic and Geometric Sequences and their Summation


14 1 sequences l.jpg

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 l.jpg

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 l.jpg

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.


P 159 ex 14a l.jpg

P.159Ex. 14A


An arithmetic sequence a s a p is a sequence having a common difference l.jpg

14.2 Arithmetic Sequence

An arithmetic sequence(A.S. /A.P.) is a sequence having a common difference.


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14.2 Arithmetic Sequence

Illustrative Examples


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14.2 Arithmetic Sequence


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14.2 Arithmetic Sequence


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14.2 Arithmetic Sequence


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P.166Ex. 14B


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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.


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14.2 Arithmetic Sequence

Arithmetic Means

Insert two arithmetic means between 11 and 35.


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14.2 Arithmetic Sequence

Insert two arithmetic means between 11 and 35.


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P.170Ex. 14C


A geometric sequence g s g p is a sequence having a common ratio l.jpg

14.3 Geometric Sequence

A geometric sequence(G.S. / G.P.) is a sequence having a common ratio.


Illustrative examples l.jpg

14.3 Geometric Sequence

Illustrative Examples


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14.3 Geometric Sequence


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14.3 Geometric Sequence


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14.3 Geometric Sequence


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P.176Ex. 14D


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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.


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14.3 Geometric Sequence

Geometric Means

Insert two geometric means between 16 and -54.


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14.3 Geometric Sequence

Insert two geometric means between 16 and -54.


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P.181Ex. 14E


Let s consider a sequence t 1 t 2 t 3 t 4 t n l.jpg

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 l.jpg

14.5 Arithmetic Series

Arithmetic Sequence : 2, 5, 8, 11, …

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


Formula of arithmetic series l.jpg

14.5 Arithmetic Series

Formula of Arithmetic Series

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

l


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14.5 Arithmetic Series

Formula of Arithmetic Series

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


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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)


Slide31 l.jpg

14.5 Arithmetic Series


P 189 ex 14f l.jpg

P.189Ex. 14F


Geometric sequence 3 9 27 81 geometric series 3 9 27 81 l.jpg

14.6 Geometric Series

Geometric Sequence : 3, 9, 27, 81, …

Geometric Series : 3 + 9 + 27 + 81


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14.6 Geometric Series

Formula of Geometric Series

S(n) = a + aR + aR2 +aR3+ …. + aRn-1


Formula of geometric series35 l.jpg

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 l.jpg

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)


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14.6 Geometric Series

Timing –1 on both numerator and denominator


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P.196Ex. 14G


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14.6 Geometric Series

Sum to Infinity of a Geometric Series


Consider such a geometric series l.jpg

14.6 Geometric Series

Sum to Infinity of a Geometric Series

Consider such a Geometric Series

What is the value of common ratio R ?


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14.6 Geometric Series

Sum to Infinity of a Geometric Series

Consider Rn where n tends to the infinity


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What will occur for if n tends to infinity ?

where –1< R <1


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Summation of a geometric Series to infinity


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P.203Ex. 14H


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(extension module)

Summation Notation


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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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