Arithmetic Series

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# Arithmetic Series - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Arithmetic Series' - xanthe

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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,….

In an arithmetic series the difference between the terms is constant.

The difference is called the common difference

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

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

2Sn = 65 + 65 + 65 + 65 + ….. + 65 + 65

2Sn = 65x 20 (because there are 20 terms)

2Sn = 1300

Sn = 650 (divide by 2)

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

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!!

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 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

### 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