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Arithmetic Series

Arithmetic Series. Understand the difference between a sequence and a series Proving the nth term rule Proving the formula to find the sum of an arithmetic series. If the terms of the sequence are added this becomes a finite series 4+7+10+13. Consider the infinite sequence 4,7,10,13,….

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Arithmetic Series

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  1. Arithmetic Series Understand the difference between a sequence and a series Proving the nth term rule Proving the formula to find the sum of an arithmetic series

  2. If the terms of the sequence are added this becomes a finite series 4+7+10+13 Consider the infinite sequence 4,7,10,13,…. In an arithmetic series the difference between the terms is constant. The difference is called the common difference

  3. Using the sequence 4, 7, 10, 13… a=1st term of the sequence d=common difference An arithmetic series is also known as an arithmetic progression (AP) 3n+1 a a+d a+2d a+3d So the nth term would be…. a + (n-1)d

  4. 58 61 Proof the the sum of an Arithmetic Series Call the sum of the terms Sn Sn= 4 + 7 + 10 + 13 + ….. + 58 + 61 Reverse the order Sn= 61+58 + 55+ 52 + ….. + 4 + 7 Add the two series together 2Sn = 65 + 65 + 65 + 65 + ….. + 65 + 65 2Sn = 65x 20 (because there are 20 terms) 2Sn = 1300 Sn = 650 (divide by 2)

  5. a=first term, d=common difference, L=last term Proof the the sum of an Arithmetic Series Sum the first n terms then reverse the order Sn= a + (a+d) + (a+2d) + (a+3d) + ….. + (L-2d) + (L-d) + L Sn= L + (L-d) + (L-2d) + (L-3d) + ….. +(a+2d) + (a+d)+ a Add the two series together 2Sn= (a+L)+(a+L)+ (a+L) + (a+L) + ….. + (a+L) + (a+L)+(a+L) 2Sn = n(a+L) (because there are n terms) Sn = n(a+L) 2 Nearly there!!

  6. a=first term, d=common difference, L=last term Sn = n(a+L) 2 Proof the the sum of an Arithmetic Series L (the last term) is also the nth term which we know has the formula a+(n-1)d so if we substitute for L in the formula above we get…. Sn = n[a+a+(n-1)d] 2 Sn = n[2a+(n-1)d] 2 You need to learn this formula

  7. EXAMPLE 1Find the sum of the first 30 terms in the series 3+9+15+… a=3, d=6, n=30 Using the formula Sn = n[2a+(n-1)d] 2 Sn = 30[2x3+(30-1)6] 2 Sn = 15[6+(29x6)] Sn = 15x180 = 2700

  8. EXAMPLE 2a)Find the nth term of the arithmetic series 7+11+15+..b)Which term of the sequence is equal to 51?c)Hence find 7+11+15+…+51 a) a=7, d=4 so the nth term is 4n+3 b) 4n+3= 51 4n = 48 (subtract 3) n = 12 (divide by 4) c) Using the formula Sn = n[2a+(n-1)d] a=7, d=4 and n=12 2 Sn = 12[2x7+(12-1)4] 2 Sn = 6[14+(11x4)] Sn = 6x58 = 348

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