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SEQUENCES AND SERIES See the following sets of numbers, do they look cumbersome? A A 1,5,9,13,17,21,25,29, ' 2 2 2 2 2 2 2 2 B A1 ,2 ,3 , 4 ,5 ,6 ,7 ,8 , ' 1 f ff 1 f 1 f 1 f 1 f 1 f ff 1f ff ff ff ff ff ' ff ff ff ff ff ff ff f f f f f f ff ff ff ff ff ff ff CA , , , , , , , 10 11 12 13 14 15 16 But look again closely. Do you see a specific order or pattern in each set? Isn’t it easy to predict any term (say the 20th) in each of the above? We note that in A we add 4 each time to get the next number, in B we take the square of the next integer and in C we take the reciprocals of the next integer. Thus the 20th term in A is 77, in B is 1f ff ff 20 and in C is ff These are called sequences, where the subsequent terms are decided 2 . 39 by a given relation (which is expressed as a function f(n) ). SEQUENCE A sequence is a function f whose domain is the set of all natural numbers (infinite sequence) or some subset of natural numbers from 1 up to some larger number (finite sequence). The values f(1), f(2), f(3). . . . are the terms of the sequence. Let the terms of a sequence be given by the function by f n = n 2 + 1 where n is a natural number A ` a f 1 = 1 + 1 = 2, f 2 = 2 + 1 = 5, f 3 = 3 + 1 = 10, ` a 2 ` a 2 ` a 2 Here f 4 = 4 + 1 = 17, f 5 = 1 + 1 = 26, ……A ` a 2 ` a 2 The sequence becomes 2, 5, 10, 17, 26, ………………A 1, 5, 9, 13, 17 is a finite sequence with 5 terms. 1, 5, 9, 13, 17, 21,……… is an infinite sequence. A sequence is represented by its range. f n = an is used to denote the range elements of ` a the function. a1 , a2 , a3 , ……an , …A are the first term, second term, third term,….nth term,…. respectively of the sequence. Example : Give first 3 terms of the sequence an = n 2 + n + 1 . Solution : Putting n = 1,2,3 in an = n 2 + n + 1, we get a1 =1 + 1 + 1 = 3, a2 = 2 + 2 + 1 = 7, a3 = 3 + 3 + 1 = 13 2 2 2 The sequence would be written 3, 7, 13,……… n fffff ffff Example : Give first 3 terms of the sequence an = fffff.ffff n +1 2 n fffff ffff ffff fffff Solution : Putting n = 1,2,3 in an = 2 , we get n +1 1 ffff 1f 2 fffff 2f 3 fffff 3f ffff f ffff ffff f ffff a1 = fffff f, a2 = fffff f, a3 = fffff ff = = ffff ff ffff ff = 1 +1 3 2 +1 5 3 + 1 10 2 2 2 1f 2f 3f f f ffff The sequence would be written as f, f, ff, AAAAAAA 3 5 10 SERIES If a1 ,a2 ,a3 ,a4 , ……AA ,an , ……… is a sequence, then the expression of their summation a1 + a2 + a3 + a4 + ……A + an + …… is a series. If we take m terms of a finite sequence, the series can be written with summation notation Σ (Sigma notation ). m X ak = a1 + a2 + a3 + a4 + ……A + am k=1 = The sum of all , the value of k being all the natural numbers from 1 to k, k is called the summation variable A 4 1 Example 3 : Write in expanded form : X fffff ffff ffff fffff k=1 k + 1 3 1 fffff ffff ffff fffff Solution : Replace k in 3 with integers from 1 to 4 and add them k +1 4 1 1 1 1 1 f f 1f 1f 1f 1f ff ff fffff ffff ffff X fffff = fffff fffff fffff fffff = ffff fffff fffff fffff ffff ffff ffff ffff ffff + 3ffff+ ffff + 3 ffff + + ff ff f f ff ff + k=1 k + 1 1 +1 2 +1 3 +1 4 +1 3 3 3 2 9 28 65 SERIES IDENTITIES: n n n n n n X ak + X bk = X ak + bk X ak @ X bk = X ak @ bk b c b c k=1 k=1 k=1 k=1 k=1 k=1 n n n X cak = c X ak X c = cn k=1 k=1 k=1 nfff+fff2nffff n f 1 ff + 1 n ` a n ` a` a n@1 nfffffff fffffff ffffff X k = fffffff fffffffffffff fffffffffffff X k = ffffffffffffff 2 k=1 2 k=1 6 n + 1 2n + 1 3n 2 + 3n @ 1 ab c n+1 n 2 ` a2 n ` a` nfffffffffffffffffffffffff nffffffff ffffffff fffffff X k = ffffffff Xk = ffffffffffffffffffffffff fffffffffffffffffffffffff fffffffffffffffffffffffff 3 4 k=1 4 k=1 30 ARITHMATIC SEQUENCE & SERIES A sequence where we start with a number a and repeatedly add a fixed constant d to get the subsequent numbers, is called an Arithmatic Sequence. If we start with 2 and add a fixed constant 3 to it repeatedly, we get the sequence 2, 5, 8, 11, 14, 17, AAAAAAAAAAAA This is an arithmatic sequence. An Arithmatic Sequence is always of the form a, a + 2d, a + 3d, a + 4d, a + 5d, AAAAAAAAAAA where we start with the number a and keep on adding d to get the subsequent terms. The number a is the first term (can be written as a1 too) and d is the common difference. To get d in a given arithmetic sequence, subtract any term from its next term. d = a2 @ a1 = a3 @ a2 = a4 @ a3 = ……AA = an @ an @ 1 = ……A term of an arithmetic sequence is given by The nth ` an = a + n @ 1 d where an = nth term, a = a1 = first term, d = common difference A a If S n is the sum of n terms of a finite arithmetic sequence, then nf a nf ff ff S n = f a + a n = f 2a + n @ 1 d ` B ` a C 2 2 Example : Write the first 5 terms of the arithmetic sequence 11,8,….. and find its 100th term. Solution : Here a1 = a = 11 and a2 = 8 A So d = a2 @ a1 = 8 @ 11 = @ 3 A Thus each term can be found by adding -3 to the previous term. Hence the first 6 terms are 11, 8, 5, 2, -1, -4. For the 100th term n = 100, and we have already found a = 11, d = @ 3 an = a + n @ 1 d ` a Using a100 = 11 + 100 @ 1 @ 3 ` a` a = 11 + 99 @ 3 ` a = 11 @ 297 = @ 286 Example : Find the sum of first 12 terms of the arithmetic sequence given by an = 2n + 5 A Solution : Putting n=1 and 2 we get, a = a1 = 2 B 1 + 5 = 7, a2 = 2 B 2 + 5 = 9 A So d = a2 @ a1 = 9 @ 7 = 2 A nf f ff Using the summation formula S n = 2a + n @ 1 d B ` a C 2 12f ff f f ff we get S 12 = 2 B7 + 12 @ 1 2 = 216 A B ` a C 2 GEOMETRIC SEQUENCE & SERIES : A sequence where we multiply each term with a non-zero constant to get the next term, is a Geometric Sequence. If we start with 2 and multiply it repeatedly by a non-zero fixed number 3, we get the 2 3 sequence 2, 2 B3, 2 B3 , 2 B3 , AAAAAAAAAAAA or 2, 6, 18, 54, AAAAAAA This is a geometric sequence. A Geometric Sequence is always of the form a, ar 2 , ar 3 , ar 4 , ar 5 AAAAAAAAAAAAAA where we start with the number a and keep on multiplying with r to get the subsequent terms. The number a is the first term (can be written as a1 too) and r is the common ratio. r is the ratio of any term to its previous term, aff aff aff anff f 3 r = ff ff ff ……AA = ffff ……A 2f f 4 = f= f= f f ffff fff = a1 a2 a3 an @ 1 Examples of Geometric sequence : 1f 1f ff ff f f 1f 1f f f ff ff 1f 1f , , ff ffAAAAAAAAA , , f a = f, r = f f 2 6 18 54 2 3 1, @ 10, 20, @ 40, 80, AAAAAAAAAA a = 1, r = @ 5 2, 0.2, 0.02, 0.002, AAAAAAAAAA a = 2, r = 0.1 The nth term of a Geometric Sequence is given by an = a r n @ 1 where an = nth term, a = a1 = first term, r = common ratio A A Geometric Series is the sum of the terms as indicated above. The sum of n terms of it @rn 1fffff fffff fffff is given by S n = a fffffwith r ≠ 1 1@r Example : Write the 6th term and sum of first 6 terms of the geometric sequence 9ff 4, 6, f, AAAAAAAAA 2 Solution : aff 6f 3f f f f f f ff f f Here in this geometric sequence, a = 4, and a2 = 6, the common ratio r = 2 = = . So, a1 4 2 f g6 @ 1 3f f f 243f using an = a r n@1 , 6th term = a6 = 4 B = fff ff

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Review of Sequences and Series.

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