Sequences & Progressions

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Sequences & Progressions

  1. 1. SEQUENCE & PROGRESSION
  2. 2. SEQUENCEA sequence is a function whose domain is the set N ofnatural numbers. Since the domain for every sequence isthe set N of natural numbers, therefore a sequence isrepresented by its range. If f : N R, then f(n) = tn n N iscalled a sequence and is denoted by{f(1), f(2), f(3),...............} = {t1, t2, t3, ......................} = {tn}
  3. 3. REAL SEQUENCEA sequence whose range is a subset of R is called a realsequence.Types of Sequence :On the basis of the number of terms there are two types ofsequence.1. Finite sequences : A sequence is said to be finite if it has finite number of terms.2. Infinite sequences : A sequence is said to be infinite if it has infinite number of terms.
  4. 4. SERIESBy adding or subtracting the terms of a sequence, we getan expression which is called a series. If a1, a2, a3,........anis a sequence, then the expression a1 + a2 + a3 + ...... + anis a series.Example : (i) 1 + 2 + 3 + 4 + .................... + n (ii) 2 + 4 + 8 + 16 + .................
  5. 5. PROGRESSIONIt is not necessary that the terms of a sequence alwaysfollow a certain pattern or they are described by someexplicit formula for the nth term. Those sequences whoseterms follow certain patterns are called progressions.An arithmetic progression (A.P.)A.P. is a sequence whose terms increase or decrease by afixed number. This fixed number is called the commondifference. If a is the first term & d the commondifference, then A.P. can be written as a, a + d, a + 2 d,.......a + (n - 1) d,........Example – 4, – 1, 2, 5 ...........
  6. 6. NTH TERM OF AN A.P.Let a be the first term and d be the common difference ofan A.P., then tn = a + (n – 1) d where d = an – an – 1THE SUM OF FIRST N TERMS OF AN A.P.
  7. 7. RTH TERM OF AN A.P.rth term of an A.P. when sum of first r terms is given is tr = Sr – Sr – 1.
  8. 8. EXAMPLES Which term of the sequence 2005, 2000, 1995, 1990, 1985, ............. contains the first negative term Ans. 403 For an A.P. show that tm + t2n + m = 2 tm + n Find the maximum sum of the A.P. 40, 38, 36, 34, 32, ... Ans. 420
  9. 9. PROPERTIES OF A.P. The common difference can be zero, positive or negative. If a, b, c are in A.P. 2 b = a + c & if a, b, c, d are in A.P. a + d = b + c. Three numbers in A.P. can be taken as a - d, a, a + d; four numbers in A.P. can be taken as a - 3d, a - d, a + d, a + 3d; five numbers in A.P. are a - 2d, a - d, a, a + d, a + 2d & six terms in A.P. are a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d etc. The sum of the terms of an A.P. equidistant from the beginning & end is constant and equal to the sum of first & last terms. Any term of an A.P. (except the first) is equal to half the sum of terms which are equidistant from it. an = 1/2 (an-k + an+k), k < n. For k = 1, an = (1/2) (an-1+ an+1); For k = 2, an = (1/2) (an-2+ an+2) and so on. If each term of an A.P. is increased, decreased, multiplied or divided by the same non zero number, then the resulting sequence is also an A.P.
  10. 10. EXAMPLES
  11. 11. ARITHMETIC MEAN (MEAN OR AVERAGE) (A.M.)
  12. 12. EXAMPLES
  13. 13. GEOMETRIC PROGRESSION (G.P.)
  14. 14. EXAMPLES
  15. 15. PROPERTIES OF G.P.
  16. 16.  If each term of a G.P. be multiplied or divided or raised to power by the some non-zero quantity, the resulting sequence is also a G.P.. If a1, a2, a3,........ and b1, b2, b3,......... are two G.P’s with common ratio r1 and r2 respectively then the sequence a1b1, a2b2, a3b3, ..... is also a G.P. with common ratio r1r2. If a1, a2, a3,..........are in G.P. where each ai > 0, then log a1, log a2, log a3,.......... are in A.P. and its converse is also true.
  17. 17. EXAMPLES
  18. 18. GEOMETRIC MEANS (MEAN PROPORTIONAL)(G.M.)
  19. 19. EXAMPLES
  20. 20. HARMONIC PROGRESSION (H.P.)
  21. 21. HARMONIC MEAN (H.M.)
  22. 22. EXAMPLES
  23. 23. RELATION BETWEEN MEANS
  24. 24. EXAMPLES
  25. 25. ARITHMETICO-GEOMETRIC SERIES
  26. 26. EXAMPLES
  27. 27. IMPORTANT RESULTS
  28. 28. EXAMPLES
  29. 29. METHOD OF DIFFERENCE
  30. 30. EXAMPLES
  31. 31. Type – 2If possible express rth term as difference of two terms as tr= f(r) – f(r ± 1).
  32. 32. NOTEIt is not always necessary that the series of first order ofdifferences i.e. u2 – u1, u3 – u2 ....... un – un–1, is alwayseither in A.P. or in G.P. in such case let u1 = T1 , u2 – u1 = T2, u3 – u2 = T3 ......., un – un–1 = Tn.So un = T1 + T2 + ..............+ Tn .........(i) un = T1 + T2 + .......+ Tn–1 + Tn .........(ii)Subtracting (ii) from (i) Tn = T1 + (T2 – T1) + (T3 – T2) + ..... + (Tn – Tn–1)Now, the series (T2 – T1) + (T3 – T2) + ..... + (Tn – Tn–1) isseries of second order of differences and when it is eitherin A.P. or in G.P. , then un = u1 + TrOtherwise in the similar way we find series of higher orderof differences and the nth term of the series.
  33. 33. EXAMPLES

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