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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