Chapter 3 sequence and series
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The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
Chapter 3 sequence and series
- 2. Introduction: The succession of quantities each of which is formed according to some definite rule or law is called a sequence. In short, sequence is set of numbers arranged in a definite order. Progression : A sequence with a constant difference or constant ratio between successive terms is called a progression eg. (i) 2, 4, 6, 8, ..................... (ii) 2, 4, 8, 16, .................... There are three types of progression. These are as follows. (i) Arithmetic progression. (A.P.) (ii) Geometric progression. (G.P.) (iii) Harmonic progression. (H.P.)
- 3. Series : A series is the algebraic sum of a sequence in which each term is connected by plus or minus e.g. (i) 1 + 2 + 3 + 4 + 5 +.................. (ii) 2 + 4 + 6 + 8 + ........................ (iii) 2 + 4 + 8 + 16 + ...................... (iv) 1 + + .................... (v) 1 – + ....................
- 4. Kinds of series : A series may be finite or infinite. 1. Finite series : A series with finite number of terms is called finite series. In this case, the last term is definite and can be determined. For example : 2 + 4 + 6 +8 +10 + 12 is a finite series. 2. Infinite series: A series with infinite number of terms is said to be an infinite series. In this case, the last term can never be determined. For example : The series of natural numbers i.e. 1+ 2 + 3 + 4 +....... is infinite.
- 5. Arithmetic Progression (Arithematic Sequence) A Sequence is said to be in arithmetic progression if the difference of any term and its preceding term is constant. In short, it is written as A. P. The constant quantity is called a common difference and is denoted by 'd'.The first term of the progression is usually denoted by 'a'. For example, (i) 3, 5, 7, .........................., here a = 3 and d = 5 – 3 = 2 (ii) 4, 2, 0, 2, 4 ................., here a = 4 and d = 2 – 4 = –2
- 6. nth term of an A. P. Let a be the first term and d be the common difference. Then The first term, t1 = a = a + (1 – 1) d The second term, t2 = a + d = a + (2 – 1) d The third term, t3 = a + 2d = a + (3 – 1) d The fourth term, t4 = a + 3d = a + (4 –1) d ................................................................... ................................................................... The nth term, tn = a + ( n –1) d Sometimes, the nth term or last term is denoted by l tn = a + ( n –1) d
- 7. Sum of n terms of the series in A. P. Let, First term = a Common difference = d Last term = l No. of terms = n Sum of n terms = Sn Since in an A. P., the successive terms increase or decrease by a constant number 'd', so the sequence in A. P. is Sn = (a + l) .........................(i) = {a + a +( n –1) d. Sn = {2a + (n –1) d} ............................. (ii)
- 8. Arithmetic mean If three numbers are in A. P. Then the middle term is called the arithmetic mean (A. M.) of the other two. Let m be the A. M. between two quantities a and b, then a, m, b are in A. P. So, by the definition of A. P. m – a = b – m or, 2m = a + b or, m = (a+b)/2 A.M. = ( a+b)/2
- 9. n arithmetic means between two numbers. When any number of terms are in A. P., all the terms between the first and the last term are called the arithmetic means of these two terms. Thus, 2, 5, 8, 11, 14, 17, 20 are in A. P., so 5, 8, 11, 14, 17 are A.M. between 2 and 20. Let m1, m2, m3 ……………… mn be n A. Ms. between a and b. So that a, m1, m2, m3, ................mn, b are in A. P. d= b – a/n + 1
- 10. A Useful Device In A. P., we denote the terms as follows: Three numbers in A. P. a – d, a , a + d Four numbers in A. P. a – 3d, a – d, a + d, a + 3d Five numbers in A. P. a – 2d, a – d, a , a + d, a + 2d and so on.