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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

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

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

an arithmetic series is also known as an arithmetic progression ap

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

proof the the sum of an arithmetic series

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)

proof the the sum of an arithmetic series1

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!!

proof the the sum of an arithmetic series2

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

example 1 find the sum of the first 30 terms in the series 3 9 15

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

slide8

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