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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


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


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


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.159Ex. 14A


14.2 Arithmetic Sequence

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


14.2 Arithmetic Sequence

Illustrative Examples


14.2 Arithmetic Sequence


14.2 Arithmetic Sequence


14.2 Arithmetic Sequence


P.166Ex. 14B


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.


14.2 Arithmetic Sequence

Arithmetic Means

Insert two arithmetic means between 11 and 35.


14.2 Arithmetic Sequence

Insert two arithmetic means between 11 and 35.


P.170Ex. 14C


14.3 Geometric Sequence

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


14.3 Geometric Sequence

Illustrative Examples


14.3 Geometric Sequence


14.3 Geometric Sequence


14.3 Geometric Sequence


P.176Ex. 14D


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.


14.3 Geometric Sequence

Geometric Means

Insert two geometric means between 16 and -54.


14.3 Geometric Sequence

Insert two geometric means between 16 and -54.


P.181Ex. 14E


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)


14.5 Arithmetic Series

Arithmetic Sequence : 2, 5, 8, 11, …

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


14.5 Arithmetic Series

Formula of Arithmetic Series

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

l


14.5 Arithmetic Series

Formula of Arithmetic Series

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


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)


14.5 Arithmetic Series


P.189Ex. 14F


14.6 Geometric Series

Geometric Sequence : 3, 9, 27, 81, …

Geometric Series : 3 + 9 + 27 + 81


14.6 Geometric Series

Formula of Geometric Series

S(n) = a + aR + aR2 +aR3+ …. + aRn-1


14.6 Geometric Series

Formula of Geometric Series

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


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)


14.6 Geometric Series

Timing –1 on both numerator and denominator


P.196Ex. 14G


14.6 Geometric Series

Sum to Infinity of 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 ?


14.6 Geometric Series

Sum to Infinity of a Geometric Series

Consider Rn where n tends to the infinity


What will occur for if n tends to infinity ?

where –1< R <1


Summation of a geometric Series to infinity


P.203Ex. 14H


(extension module)

Summation Notation


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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