Arithmetic Progression

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Arithmetic Progression

  1. 1. Arithmetic Progression T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
  2. 2. Arithmetic Progression a) 5, 8, 11, 14, 17, 20, … 3n+2, … b) -4, 1, 6, 11, 16, … 5n – 9, . . . c) 11, 7, 3, -1, -5, … -4n + 15, . . . In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression ( AP ).  So,an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.  This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero © iTutor. 2000-2013. All Rights Reserved
  3. 3. nth term of arithmetic sequence Tn = a + d(n – 1) a = First term d = common difference n = number of terms. Common difference = the difference between two consecutive terms in a sequence. d = Tn – Tn-1 Example Find the nth term of the following AP. © iTutor. 2000-2013. All Rights Reserved
  4. 4. Finding the 956th term 56, 140, 124, 108, . . . Tn = a + d(n – 1) T956 = 156 + -16(956 – 1) T956 = 156 - 16(955) T956 = 156 - 15280 T956 = -15124 a1 = 156 d = -16 n = 956 Example Finding the number of terms in the AP 10, 8, 6, 4, 2, . . .-24 Tn = a + d(n – 1) -24 = 10 -2(n – 1) -34 = -2(n – 1) 17 = n-1 n = 18 a = 10 d = -2 Tn = -24 © iTutor. 2000-2013. All Rights Reserved
  5. 5. The 5th term of an AP is 13 and the 13th term is -19. Find the first term & the common difference. T5 = a + 4d = 13……..(1) T13= a + 12d = -19……….(2) (2) – (1): 8d = -19 - 13 8d = - 32 d = -4 Substitute d = -4 into (1): a + 4(-4) = 13 a – 16 = 13 a = 29© iTutor. 2000-2013. All Rights Reserved
  6. 6. Sn = a1 + (a1 + d) + (a1 + 2d) + …+ an Sn = an + (an - d) + (an - 2d) + …+ a1 2 )( 1 1 n n i in aan aS + == ∑= )(2 1 nn aanS += )(...)()()(2 1111 nnnnn aaaaaaaaS ++++++++= Sum of First terms of an AP © iTutor. 2000-2013. All Rights Reserved
  7. 7. 1 + 4 + 7 + 10 + 13 + 16 + 19 a1 = 1 an = 19 n = 7 2 )( 1 n n aan S + = 2 )191(7 + =nS 2 )20(7 =nS 70=nS Example Find the sum of the integers from 1 to 100 a1 = 1 an = 100 n = 100 2 )( 1 n n aan S + = 2 )1001(100 + =nS 2 )101(100 =nS 5050=nS © iTutor. 2000-2013. All Rights Reserved
  8. 8. Find the sum of the multiples of 3 between 9 and 1344 a1 = 9 an = 1344 d = 3 2 )( 1 n n aan S + = 2 )13449( + = n Sn 2 )1353(446 =nS 301719=nS )1(1 −+= ndaan )1(391344 −+= n 3391344 −+= n 631344 += n n31338 = n=446 Sn = 9 + 12 + 15 + . . . + 1344 © iTutor. 2000-2013. All Rights Reserved
  9. 9. Find the sum of the multiples of 7 between 25 and 989 a1 = 28 an = 987 d = 7 2 )( 1 n n aan S + = 2 )98728( + = n Sn 2 )1015(138 =nS 70035=nS )1(1 −+= ndaan )1(728987 −+= n 7728987 −+= n 217987 += n n7966 = n=138 Sn = 28 + 35 + 42 + . . . + 987 © iTutor. 2000-2013. All Rights Reserved
  10. 10. Evaluate a1 = 16 an = 82 d = 3 n = 23 2 )( 1 n n aan S + = 2 )8216(23 + =nS 2 )98(23 =nS 1127=nS Sn = 16 + 19 + 22 + . . . + 82 ∑= + 25 3 )73( i i © iTutor. 2000-2013. All Rights Reserved
  11. 11. Review -- Arithmetic nth term Sum of n terms )1(1 −+= ndaan 2 )( 1 n n aan S + = © iTutor. 2000-2013. All Rights Reserved
  12. 12. The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit

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