Basic numeracy-sequences-and-series

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Basic numeracy, sequences and series

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Basic numeracy-sequences-and-series

  1. 1. Click Here to Join Online Coaching Click Here www.upscportal.com Basic Numeracy
  2. 2. www.upscportal.com Click Here to Join Online Coaching Click Here Sequences & Series Arithmetic Progression (AP) An arithmetic progression is a sequence in which terms increase or decrease by a constant number called the common difference. (i) The sequence 2, 6, 10, 14, 18, 22… is an arithmetic progression whose first term is 2 and common difference 4. (ii) The sequence 2, 5/2, 3 ,7/2 ,4 …is an arithmetic progression whose first term is 2 and common difference ½. An arithmetic progression is represented by a,(a + d), (a + 2d), (a + 3d) a + (n – 1)d Here, a = first term d = common difference n = number of terms in the progression
  3. 3. www.upscportal.com Click Here to Join Online Coaching Click Here • The general term of an arithmetic progression is given by Tn = a + (n - 1) d. • The sum of n terms of an arithmetic progression is given by S, = n/2 [2a + (n – 1) d] or Sn = 2 [a + l] where l is the last term of arithmetic progression. • If three numbers are in arithmetic progression, the middle number is called the arithmetic mean of the other two terms. • If a, b, c are in arithmetic progression, then b = a+c/2 where b is the arithmetic mean. • Similarly, if ‘n’ terms al, a2, a3… an are in AP, then the arithmetic mean of these ‘n’ terms is given by AM = • If the same quantity is added or multiplied to each term of an AP, then the resulting series is also an AP.
  4. 4. www.upscportal.com Click Here to Join Online Coaching Click Here • If three terms are in AP, then they can be taken as (a – d), a, (a + d). • If four terms are in AP, then they can be taken as (a – 3d), (a – d), (a + d), (a + 3d). • If five terms are in AP, then they can be taken as (a – 2d), (a – d), a, (a + d), (a + 2d). Geometric Progression (GP) A geometric progression is a sequence in which terms increase or decrease by a constant ratio called the common ratio. (i) The sequence 1, 3, 9, 27, 81…is a geometric progression whose first term is 1 and common ratio 3. (ii) The sequence is a geometric progression whose first term is 1 and common ratio 1/3.
  5. 5. www.upscportal.com Click Here to Join Online Coaching Click Here A geometric progression is represented by a, ar, ar2…arn–1. Here, a = first term r = common ratio n = number of terms in the progression. • The general term of a geometric progression is given by Tn = an–1 • The sum to n terms of a geometric progression is given by when r < 1 when r > 1 • If three numbers are in geometric progression, the middle number is called the geometric mean of the other two terms. • If a, b, c are in geometric progression, then where b is the geometric mean.
  6. 6. www.upscportal.com Click Here to Join Online Coaching Click Here • Similarly, if n terms a1, a2, a3, a4,…an are in geometric progression, then the geometric mean of 1 these n terms is given by GM = • For a decreasing geometric progression the sum to infinite number of terms is where a = first term and | r | < 1. • If every term of a GP is multiplied by a fixed real number, then the resulting series is also a GP. • If every term of a GP is raised to the same power, then the resulting series is also a GP. • The reciprocals of the terms of a GP is also a GP. • If three numbers are in GP, then they can be taken as a/r , a, ar.
  7. 7. www.upscportal.com Click Here to Join Online Coaching Click Here • If four numbers are in GP, then they can be taken as • If five numbers are in GP, then they can be taken as Harmonic Progression (HP) If the reciprocals of the terms of a series form an arithmetic progression, then the series is called a harmonic progression. (i) The sequence 4/3, 3/2, 12/7, …. is a harmonic progression as 3/4, 2/3, 7/12 is in arithmetic progression. • If a, b, c are in harmonic progression, then b = 2ac / a+c where b is the harmonic mean. Sum of Natural Series • The sum of the first n natural numbers = n (n+1) / 2 .
  8. 8. www.upscportal.com Click Here to Join Online Coaching Click Here • The sum of the square of the first n natural numbers = n (n+1) (2n+1) / 6 • The sum of the cubes of the first n natural numbers = • The sum of first n even numbers = n(n + 1) • The sum of first n odd numbers = n2 Example 1: Find the nth term and the fifteenth term of the arithmetic progression 3, 9, 15, 21… Solution. In the given AP we have a = 3, d = (9 – 3) = 6 Tn = a + (n – 1)d = 3 + (n – 1)6 = 6n – 3 T15 = (6 × 15 – 3) = 87 Example 2: Find the 10th term of the AP 13, 8, 3, –2,… Solution. In the given AP, we have a = 13, d = (8 –13) = –5 Tn = a + (n – 1)d = 13 + (n – 1)(–5) = 18 – 5n T10 = 18 – 5 (10) = –32
  9. 9. www.upscportal.com Click Here to Join Online Coaching Click Here Example 3: The first term of an AP is -1 and the common difference is -3, the 12th term is Solution. T1 = a = –1, d = –3 Tn = a + (n – 1)d = –1 + (n – 1)(–3) = 2 – 3n T12 = 2 – 3 × 12 = –34 Example 4: Which term of the AP 10, 8, 6, 4… is –28? Solution. We have, a = 10,d = (8 – 10) = –2, Tn = –28 Tn = a + (n – 1)d – 28 = 10 + (n – 1)(–2) = n = 20 Example 5: The 8th term of an AP is 17 and the 19th term is 39. Find the 20th term. Solution. T8 = a + 7d =17 ...(i) T19 = a + 18d = 39 ... (ii) On subtracting Eq. (i) from Eq. (ii), we get 11d = 22 = d = 2
  10. 10. www.upscportal.com Click Here to Join Online Coaching Click Here Putting d = 2 in Eq. (i), we get a + 7(2) = 17 a = (17 – 14) = 3 . . . First term = 3, Common difference = 2 T20= a + 19d = 3 + 19(2) = 41
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