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Arithmetic and Geometric Sequences and their Summation 14.1 Sequences Find the next two terms of the following sequences : (1) 2, 5, 8, 11,…… arithmetic sequence (2) 2, 6, 18, 54, …. geometric sequence (3) 2, 4, 8, 16,……. geometric sequence (4) 5, -25, 125, -625, …. geometric sequence (5) 3, 4, 6, 9, 13, ……. (6) 5, 2, -1, -4, ….. arithmetic sequence (7) 0, sin20o, 2sin30o, 3sin40o 14.1 Sequences Consider the following sequence: 1, 3, 5, 7, 9, ….., 111 1 is the first term of the sequence,mathematically, T(1) = 1 or T1 = 1 3 is the second term of the sequence, mathematically, T(2) = 3 or T2 = 3 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, …. The sequence is formed from timing 2 to the previous term. So, the sequence can be represented by the general term T(n) = 2n or Tn = 2n P.159 Ex. 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.166 Ex. 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. ac b 2 14.2 Arithmetic Sequence Arithmetic Means Insert two arithmetic means between 11 and 35. T (1) a 11.......... 1) ....( T (2) a d .......... 2) .....( T (3) a 2d .......... 3) ...( T (4) a 3d 35.....(4) 14.2 Arithmetic Sequence Insert two arithmetic means between 11 and 35. (4) (1) 3d 24 d 8 1st arithmetic mean a d 19 2nd arithmetic mean a 2d 27 P.170 Ex. 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.176 Ex. 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. y xz 14.3 Geometric Sequence Geometric Means Insert two geometric means between 16 and -54. T (1) a 16.......... 1) ....( T (2) aR.......... 2 .........( ) T (3) aR .......... 2 3 .......( ) T (4) aR 54......( ) 3 4 14.3 Geometric Sequence Insert two geometric means between 16 and -54. ( 4) 54 27 R 3 (1) 16 8 3 R 2 1st goemetric mean aR 24 2nd goemetric mean aR 36 2 P.181 Ex. 14E 14.4 Series Let’s consider a sequence : T(1), T(2), T(3), T(4), …., T(n) 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). S (n) T (1) T (2) T (3) .... 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) n( a l ) S ( n) 2 14.5 Arithmetic Series n( a l ) S ( n) 2 n S (n) [a a (n 1)d ] 2 n S (n) [2a (n 1)d ] 2 P.189 Ex. 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 + aR + 2 aR 3 + aR4 + …. + aRn Subtracting two series S(n) = a + aR + aR2 + aR3 + …. + aRn-1 R.S(n) = aR + aR2 + aR3 + aR4 + …. + aRn S(n) –R.S(n) = a - aRn (1 – R) S(n) = a (1 – Rn) a(1 R ) n S ( n) , where R 1 1 R 14.6 Geometric Series a(1 R )n S ( n) , where R 1 1 R Timing –1 on both numerator and denominator a( R 1) n S ( n) , where R 1 R 1 P.196 Ex. 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 1 1 1 1 ..... 2 4 8 16 What is the value of common ratio R ? 1 R 2 14.6 Geometric Series Sum to Infinity of a Geometric Series 1 R 2 Consider Rn where n tends to the infinity 1 2 1 3 1 4 ( ) 0.25 ( ) 0.125 ( ) 0.0625 2 2 2 1 5 1 6 1 7 ( ) 0.03125 ( ) 0.015625 ( ) 0.0078125 2 2 2 1 What will occur for ( ) if n tends 2 to infinity ? 1 ( ) 0 2 R 0 where –1< R <1 Summation of a geometric Series to infinity a(1 R ) S ( ) , where 1 R 1 1 R a S ( ) , where 1 R 1 1 R P.203 Ex. 14H (extension module) Summation Notation 4 Consider the symbol T (r ) r 1 where T( r ) = 3r + 5 4 T (r ) r 1 = 3(1) + 5 + 3(2) + 5 +3(3) + 5 + 3(4) +5 = 50