Sequences and Series Session MPTCP04

1.Finite and infinite sequences 2.Arithmetic Progression (A.P.) - definition, n th term 3.Sum of n terms of an A.P. 4.Arithmetic Mean (A.M.) and insertion of n A.M.s between two given numbers. 5.Geometric Progression (G.P.) - definition, n th term 6.Sum of n terms of a G.P. Session Objectives

Sequence – a Definition A sequence is a function whose domain is the set N of natural numbers. _I001 a 1, a 2, a 3,..., a n,...

Finite and Infinite Sequences Finite sequence : a  a 1, a 2, a 3,..., a n Infinite sequence : a  a 1, a 2, a 3,..., a n,...  _I001

Series – a Definition If a 1, a 2, a 3,..., a n,... is a sequence, _I001 the expression a 1 +a 2 +a a n +... is called a series.

Arithmetic Progression A sequence is called an arithmetic progression (A.P.) if the difference between any term and the previous term is constant. The constant difference, generally denoted by d is called the common difference. _I002 a 1 = a a 2 = a+d a 3 = a+d+d= a+2d a 4 = a+d+d+d= a+3d First term General Term a n = a+d+d+d+...= a+(n-1)d

Is a Given Sequence an A.P.? Algorithm to determine whether a given sequence is an A.P. : Step IObtain general term a n Step IIDetermine a n+1 by replacing n by n+1 in the general term Step IIIFind a n+1 -a n. If this is independent of n, the given sequence is an A.P. _I002

Problem Solving Tip Choose Well!!!! #TermsCommon diff. 3a-d, a, a+d d 4a-3d, a-d, a+d, a+3d2d 5a-2d, a-d, a, a+d, a+2d d 6a-5d, a-3d, a-d, a+d, a+3d, a+5d2d _I002

Illustrative problem Q. If sum of three numbers in A.P. is 45, and the second number is thrice the first number, find the three numbers. _I002 A. Let the numbers be a-d, a, a+d Given that (a-d)+a+(a+d) = 45  3a = 45  a = 15 Also, a = 3(a-d)  3d = 30  d = 10  the three numbers are 5, 15, 25

Important Properties of A.P.s If A  a, a+d,..., a+(n-1)d adding constant k to each term, we get, A’  a+k, a+d+k,..., a+(n-1)d+k A’ is also an A.P. with the same common difference. _I002

Important Properties of A.P.s If A  a, a+d,..., a+(n-1)d multiplying each term by non-zero constant k, A’  ak, ak+dk,..., ak+(n-1)dk A’ is also an A.P. with common difference dk _I002

Important Properties of A.P.s a k +a n-(k-1) = a 1 +a n  k = 2, 3, 4,... (n-1) Example : Consider A  2, 4, 6, 8, 10, 12, 14, 16, 18, 20 _I002

Important Properties of A.P.s Example : Consider A  2, 4, 6, 8, 10, 12, 14, 16, 18, 20 a 3 +a 8 = 22 _I002 a k +a n-(k-1) = a 1 +a n  k = 2, 3, 4,... (n-1)

Important Properties of A.P.S Example : Consider A  2, 4, 6, 8, 10, 12, 14, 16, 18, 20 a 3 +a 8 = 22= a 5 +a 6 = 22 _I002 a k +a n-(k-1) = a 1 +a n  k = 2, 3, 4,... (n-1)

Important Properties of A.P.s Example : Consider A  2, 4, 6, 8, 10, 12, 14, 16, 18, 20 a 3 +a 8 = 22= a 5 +a 6 = 22= a 1 +a 10 = 22 _I002 a k +a n-(k-1) = a 1 +a n  k = 2, 3, 4,... (n-1)

Important Properties of A.P.s a, b, c are in A.P.  2b = a+c _I002

Important Properties of A.P.s A sequence is an A.P.  a n = An+B, A, B are constants. A is the common difference. Proof : a n = a+(n-1)d or, a n = dn+(a-d) or, a n = An+B, where A is the common difference _I002

Important Properties of A.P.s If A  a, a+d,..., a+(n-1)d take every third term, A’  a, a+3d, a+6d, A’ is also an A.P. _I002

Sum of n Terms of an A.P. S n = a 1 +(a 1 +d)+...+{a 1 +(n-2)d}+{a 1 +(n-1)d} Also, S n = {a 1 +(n-1)d}+{a 1 +(n-2)d}+{a 1 +d}+...+a 1 Adding, 2S n = n{2a 1 +(n-1)d} _I003

Sum of n Terms of an A.P. This can also be written as : _I003

Property of Sum of n Terms of an A.P. A sequence is an A.P.  S n = An 2 +Bn, where A, B are constants. 2A is the common difference. We know that,Rearranging, Or, S n = An 2 +Bn. _I003

Arithmetic Mean A is the A.M. of two numbers a and b  a, A, and b are in A.P.  A-a = b-A  2A = a+b _I004

Arithmetic Mean – a Definition then A 1, A 2, A 3,..., A n are called arithmetic means (A.M.s) of a and b A1A1 A2A2 A3A3 A4A4 A5A a 6 b If n terms A 1, A 2, A 3,... A n are inserted between two numbers a and b such that a, A 1, A 2, A 3,..., A n, b form an A.P., _I004

Arithmetic Mean – Common Difference Let n A.M.s be inserted between two numbers a and b Let the common difference be d Now b = a+(n+2-1)d = a+(n+1)d A1A1 A2A2 A3A3 A4A4 A5A a 6 b The A.P. thus formed will have (n+2) terms. _I004

Property of A.M.s Let n A.M.s A 1, A 2, A 3,..., A n be inserted between a and b. Then, _I004

Illustrative Problem Q.Insert 3 A.M.s between -4 and 3 A.Let the required A.M.s be A 1, A 2 and A 3. Common difference d = _I004

Geometric Progression Consider a family where every female of each generation has exactly 2 daughters. It is then possible to determine the number of females in each generation if the generation number is known. 1 st Generation1 female 2 nd Generation2 females 3 rd Generation4 females Such a progression is a Geometric Progression (G.P.) _I005

Geometric Progression A sequence is called a geometric progression (G.P.) if the ratio between any term and the previous term is constant. _I005 The constant ratio, generally denoted by r is called the common ratio. a 1 = a a 2 = ar a 3 = ar 2 a 4 = ar 3 a n = ar (n- 1) First term General Term

Problem Solving Tip Choose Well!!!! #TermsCommon ratio 3a/r, a, arr 4a/r 3, a/r, ar, ar 3 r 2 5a/r 2, a/r, a, ar, ar 2 r 6a/r 5, a/r 3, a/r, ar, ar 3, ar 5 r 2 _I005

Important Properties of G.P.s If G  a, ar, ar 2,..., ar n-1 multiplying each term by non- zero constant k, G’  ka, kar, kar 2,..., kar n-1 G’ is also a G.P. with the same common ratio. _I005

Important Properties of G.P.s If G  a, ar, ar 2,..., ar n-1 taking reciprocal of each term, G’ is also a G.P. with a reciprocal common ratio. _I005

Important Properties of G.P.s If G  a, ar, ar 2,..., ar n-1 raising each term to power k, G’  a k, a k r k, a k r 2k,..., a k r (n-1)k G’ is also a G.P. with common ratio r k. _I005

Important Properties of G.P.s a k a n-(k-1) = a 1 a n  k = 2, 3, 4,... (n-1) Example : Consider G  1, 2, 4, 8, 16, 32, 64, 128, 256, 512 _I005

Important Properties of G.P.s Example : Consider G  1, 2, 4, 8, 16, 32, 64, 128, 256, 512 a 3 a 8 = 512 _I005 a k a n-(k-1) = a 1 a n  k = 2, 3, 4,... (n-1)

Important Properties of G.P.s Example : Consider G  1, 2, 4, 8, 16, 32, 64, 128, 256, 512 a 3 a 8 = 512= a 5 a 6 = 512 _I005 a k a n-(k-1) = a 1 a n  k = 2, 3, 4,... (n-1)

Important Properties of G.P.s Example : Consider G  1, 2, 4, 8, 16, 32, 64, 128, 256, 512 a 3 a 8 = 512= a 5 a 6 = 512= a 1 a 10 = 512 _I005 a k a n-(k-1) = a 1 a n  k = 2, 3, 4,... (n-1)

Important Properties of G.P.s a, b, c are in G.P.  b 2 = ac _I005

Important Properties of G.P.s If G  a, ar, ar 2,..., ar n-1 take every third term, G’  a, ar 3, ar 6,... G’ is also a G.P. _I005

Important Properties of G.P.s a 1, a 2, a 3,..., a n is a G.P. of positive terms  loga 1, loga 2, loga 3,... loga n is an A.P. _I005

Sum of n Terms of a G.P. S n = a+ar+ar 2 +ar ar (n-1) ………(i) Multiplying by r, we get, rS n = ar+ar 2 +ar ar (n-1) +ar n ……...(ii) Subtracting (i) from (ii), (r-1)S n = a(r n -1) _I006

Class Exercise Q1. Q. If log 2, log (2 x -1) and log (2 x +3) are in A.P., find x. _I002

Class Exercise Q1. Q. If log 2, log (2 x -1) and log (2 x +3) are in A.P., find x. _I002 A. Given that log(2 x -1)-log2 = log(2 x +3)-log(2x-1)

Class Exercise Q2. Q. Show that there is no infinite A.P. which consists only of distinct primes. _I002

Class Exercise Q2. _I002 A. Let, if possible, there be an A.P. consisting only of distinct primes : a 1, a 2, a 3,..., a n,... a n = a 1 +(n-1)d Q. Show that there is no infinite A.P. which consists only of distinct primes. Thus, (a 1 +1) th term is a multiple of a 1. Thus, no such A.P. is possible. Q.E.D.

Class Exercise Q3. _I003 Q., where S n denotes the sum of the first n terms of an A.P., then common difference is : (a) P+Q(b) 2P+3Q (c) 2Q(d) Q (J.E.E. West Bengal 1994)

Class Exercise Q3. _I003 Q., where S n denotes the sum of the first n terms of an A.P., then common difference is : (a) P+Q(b) 2P+3Q (c) 2Q(d) Q (J.E.E. West Bengal 1994) A. a n = S n - S n-1  Ans : (d).

Class Exercise Q4. _I003 Q. If 12 th term of an A.P. is -13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

Class Exercise Q4. _I003 Q. If 12 th term of an A.P. is -13 and the sum of the first four terms is 24, what is the sum of first 10 terms? A. Given that, a 12 = a 1 +11d = (i) S 4 = 2(2a 1 +3d) = (ii) Solving (i) and (ii) simultaneously, we get, a 1 = 9, d = -2  S 10 = 5(2a 1 +9d) = 5(18-18) = 0

Class Exercise Q5. _I004 Q. Find the value of n so that be an A.M. between a and b (a, b are positive).

Class Exercise Q5. A. Given that, _I004 Q. Find the value of n so that be an A.M. between a and b (a, b are positive). Dividing throughout by b n+1, we get,

Class Exercise Q5. _I004 Q. Find the value of n so that be an A.M. between a and b (a, b are positive).  n = 0

Class Exercise Q6. _I004 Q. 53 A.M.s are inserted between 2 and 98. Find the 27 th A.M.

Class Exercise Q6. _I004 Q. 53 A.M.s are inserted between 2 and 98. Find the 27 th A.M. A. Common difference

Class Exercise Q7. _I005 Q. If the 3 rd term of a G.P. is 4, what is the product of the first five terms?

Class Exercise Q7. _I005 Q. If the 3 rd term of a G.P. is 4, what is the product of the first five terms? A. Let the first 5 terms of the G.P. be : Required product= a 5 = (a 3 ) 5 =4 5

Class Exercise Q8. _I005 Q. If the 4 th, 7 th, 10 th term of a G.P. are p, q, r respectively, then (a) p 2 = q 2 +r 2 (b) q 2 = pr (c) p 2 = qr(d) pqr+pq+q = 0 (M.N.R. 1995)

Class Exercise Q8. _I005 Q. If the 4 th, 7 th, 10 th term of a G.P. are p, q, r respectively, then (a) p 2 = q 2 +r 2 (b) q 2 = pr (c) p 2 = qr(d) pqr+pq+q = 0 (M.N.R. 1995) A. Let the first term of the G.P. be  and common ratio be .  Ans : (b)  p =  3, q =  6, r =  9 Now, pr=  2  12 = ( 6 ) 2 = q 2

Class Exercise Q9. _I006 Q. Find the sum to n terms of the sequence 6, 66, 666,...

Class Exercise Q9. _I006 Q. Find the sum to n terms of the sequence 6, 66, 666,... A. S n = ( [n terms])

Class Exercise Q10. _I006 Q. How many terms of the G.P. are needed to give the sum ?

Class Exercise Q10. _I006 Q. How many terms of the G.P. are needed to give the sum ? A. Common ratio Let the required number of terms be n.

Class Exercise Q10. _I006 Q. How many terms of the G.P. are needed to give the sum ?  n = 5

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