Sequences and Series


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1 Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO
2 While the vst mjority of Euclid questions in this topic re use formule for rithmetic or geometric sequences, we will lso include few involving summtions nd different types of sequences. TOOLKIT Arithmetic Sequences Description Generl k th term Sum of n terms Spcing of terms Sequences with common difference t k = + (k 1)d where is the first term nd d is this common difference S n = n ( + t n) = n ( + (n 1)d) Becuse of the equl spcing of terms we hve t k + t l = t m + t n if nd only if k + l = m + n Geometric Sequences Description Sequences with common rtio Generl k th term t k = r k 1 where is the first term nd r is the rtio Sum of n terms S n = (1 rn ) (1 r) Spcing of terms Becuse of the equl spcing of terms we hve t k t l = t m t n if nd only if k + l = m + n Infinite sum If the rtio r stisfies the condition r < 1, we cn dd n infinite number of terms using S = 1 r Other Of course rithmetic nd geometric sequences re smll subset of ll sequences, even though they re emphsized in high school mthemtics. Some extensions tht frequently pper on contests often involve: First n integers First n squres First n cubes Telescoping series n k = n(n + 1) n k n(n + 1)(n + 1) = 6 ( ) n k 3 n(n + 1) = If t k = u k u k 1 then n n t k = (u k u k 1 ) = u n u 0 CENTRE FOR EDUCATIONS IN MATHEMATICS COMPUTING
3 SAMPLE PROBLEMS 1. Wht is the sum of ll multiples of 7 or 11 less thn 1000? Solution Since we re dding ( ) + ( ), we re dding two rithmetic sequences. However the multiples of 77 re included in both sequences nd so must be subtrcted (in order to void counting them twice) fter we dd the sequences bove. Therefore the required sum is ( ) + ( ) ( ). Now since 994 is the 14 nd term in the first sequence we hve the sum of the first sequence is 14 ( ). Thus the required sum is 14 ( ) + 90 ( ) 1 ( ) = ( )(1001) = (110)(1001) = A sequence is given such tht t 1 = 1 nd t n+1 = t n + 3n + 3n + 1. Evlute t 100. Solution Since the difference, t n t n 1 is not constnt, the series is not rithmetic. Now setting n = 1 we find t = = 8. Setting n = we find t 3 = = 7. These fcts suggest t n = n 3 for every n. To prove tht t n = n 3 is n lternte definition for the sme sequence, we first note tht t 1 = 1 = 1 3. Further, consider two djcent terms in the sequence given by the lternte definition, i.e. t n = n 3 nd t n+1 = (n + 1) 3. Then the difference between these terms is t n+1 t n = (n + 1) 3 (n) 3 = (n 3 + 3n + 3n + 1) n 3 = 3n + 3n + 1 t n+1 = t n + 3n + 3n + 1 which mtches the originl definition of the sequence. We hve proved tht the originl sequence cn be expressed s t n = n 3, nd thus t 100 = CENTRE FOR EDUCATIONS IN MATHEMATICS COMPUTING 3
4 3. If, b, + b, nd b re positive numbers tht form 4 consecutive terms in geometric sequence, find. Solution The rtios of successive terms will be equl since we hve geometric sequence. So Therefore, b = b + b = + b b ( ) ( ) b + b = b b b = 0 ( ) b 1 = 0 ( ) b = where we hve chosen the positive root since nd b re positive. Also from (*), b = + b b = + b = 1 + b = CENTRE FOR EDUCATIONS IN MATHEMATICS COMPUTING 4
5 PROBLEM SET 1. In geometric series, t 5 + t 7 = 1500 nd t 11 + t 13 = Find ll possible vlues for the first three terms.. Given tht, b nd c re successive terms in n rithmetic sequence clculte x if (b c)x + (c )x + ( b) = If x, 4, y re successive terms in n rithmetic sequence nd x, 3, y re successive terms in geometric sequence, clculte 1 x + 1 y. 4. Three different numbers, whose product is 15, re 3 consecutive terms in geometric sequence. At the sme time they re the first, third nd sixth terms of n rithmetic sequence. Find the numbers. k(k + 1) 5. The kth tringulr number is given by T k = k = = k + k. The first few tringulr numbers re 1, 3, 6, 10, 15, 1. Find the sum of the first 00 tringulr numbers. 6. If the interior ngles of pentgon form n rithmetic sequence nd one interior ngle is 90 o, find ll possible vlues of the lrgest ngle in the pentgon. 7. Find the 4 integers, b, c nd d tht stisfy the following conditions: the sum of b nd c is 30 the sum of nd d is 35 the numbers < b < c < d re in geometric sequence the sum of the squres of the 4 numbers is A sequence t 1, t, t 3 is formed by choosing t 1 t rndom from the set {1,, 3}, t t rndom from the set {4, 5, 6}, nd t 3 t rndom from the set {7, 8, 9}. Wht is the probbility tht t 1, t, t 3 is n rithmetic sequence? 9. The sum of 5 consecutive integers is 500. Determine the smllest of the 5 integers. 10. Wht is the number of terms in the rithmetic sequence 1994, 199, 1990,..., 199, 1994? 11. The sum of the first n terms of sequence is S n = 3 n 1, where n is positive integer. () If t n represents the nth term of the sequence, determine t 1, t, t 3. (b) Prove tht t n+1 t n is constnt for ll vlues of n. 1. How mny terms in the rithmetic sequence 7, 14, 1,... re between 40 nd 8 001? 13. If f is function such tht f(1) = nd f(n+1) = 3f(n) for n = 1,, 3,..., wht is the vlue of f(100)? 14. For the fmily of lines with equtions of the form px + qy = r, nd which ll pss through the point ( 1, ), prove tht p, q, nd r re consecutive terms of n rithmetic sequence. 15. An rithmetic sequence S hs terms t 1, t, t 3,..., where t 1 = nd the common difference is d. The terms t 5, t 9, nd t 16 form threeterm geometric sequence with common rtio r. Prove tht S contins n infinite number of threeterm geometric sequences, ll hving the sme common rtio r. 16. In the sequence 5, 3,, 5,...,, ech term fter the first two is constructed by tking the preceding term nd subtrcting the term before it. Wht is the sum of the first 3 terms in the sequence? CENTRE FOR EDUCATIONS IN MATHEMATICS COMPUTING 5
6 17. Consider the sequence t 1 = 1, t = 1 nd t n = ( ) n 3 t n where n 3. Wht is the vlue of t 1998? n The nth term of n rithmetic sequence is given by t n = 555 7n. If S n = t 1 + t t n, determine the smllest vlue of n for which S n < 0. CENTRE FOR EDUCATIONS IN MATHEMATICS COMPUTING 6
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