Arithmetic Series Understand the difference between a sequence

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

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

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

Proof the sum of an Arithmetic Series n 1 2 3 4 …. . 3 n+1 4 7 10 13 …. . 19 58 20 61 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 2 Sn = 65 + …. . + 65 2 Sn = 65 x 20 (because there are 20 terms) 2 Sn = 1300 Sn = 650 (divide by 2)

Proof the sum of an Arithmetic Series a=first term, d=common difference, L=last term n 1 2 3 4 …. . n-1 n a a+d a+2 d a+3 d …. . L-d L Sum the first n terms then reverse the order Sn= a + (a+d) + (a+2 d) + (a+3 d) + …. . + (L-2 d) + (L-d) + L Sn= L + (L-d) + (L-2 d) + (L-3 d) + …. . +(a+2 d) + (a+d)+ a Add the two series together 2 Sn= (a+L)+ (a+L) + …. . + (a+L)+(a+L) 2 Sn = n(a+L) (because there are n terms) Sn = n(a+L) 2 Nearly there!!

Proof the sum of an Arithmetic Series a=first term, d=common difference, L=last term Sn = n(a+L) 2 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[2 a+(n-1)d] 2 You need to learn this formula

EXAMPLE 1 Find the sum of the first 30 terms in the series 3+9+15+… a=3, d=6, n=30 Using the formula Sn = n[2 a+(n-1)d] 2 Sn = 30[2 x 3+(30 -1)6] 2 Sn = 15[6+(29 x 6)] Sn = 15 x 180 = 2700

EXAMPLE 2 a)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 4 n+3 b) 4 n+3= 51 4 n = 48 (subtract 3) n = 12 (divide by 4) c) Using the formula Sn = n[2 a+(n-1)d] a=7, d=4 and n=12 2 Sn = 12[2 x 7+(12 -1)4] 2 Sn = 6[14+(11 x 4)] Sn = 6 x 58 = 348