Arithmetic Series

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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

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Consider the infinite sequence 4,7,10,13,.
If the terms of the sequence are added this
becomes a finite series 471013
In an arithmetic series the difference between
the terms is constant.
The difference is called the common difference
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An arithmetic series is also known as an
arithmetic progression (AP)
Using the sequence 4, 7, 10, 13 a1st term of
the sequence dcommon difference
n 1 2 3 4
4 7 10 13

3n1
a
ad
a2d
a3d
So the nth term would be.
a (n-1)d
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Proof the the sum of an Arithmetic Series
n 1 2 3 4 .. 19 20
3n1 4 7 10 13 ..
61
58
Call the sum of the terms Sn
Sn 4 7 10 13 .. 58 61
Reverse the order
Sn 6158 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)
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Proof the the sum of an Arithmetic Series
afirst term, dcommon difference, Llast term
n 1 2 3 4 .. n-1 n
a ad a2d a3d .. L-d L
Sum the first n terms then reverse the order
Sn a (ad) (a2d) (a3d) .. (L-2d)
(L-d) L
Sn L (L-d) (L-2d) (L-3d) .. (a2d)
(ad) a
Add the two series together
2Sn (aL)(aL) (aL) (aL) .. (aL)
(aL)(aL)
2Sn n(aL) (because there are n terms)
Sn n(aL) 2
Nearly there!!
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Proof the the sum of an Arithmetic Series
afirst term, dcommon difference, Llast term
Sn n(aL) 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 naa(n-1)d 2
Sn n2a(n-1)d 2
You need to learn this formula
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EXAMPLE 1Find the sum of the first 30 terms in
the series 3915
a3, d6, n30
Using the formula Sn n2a(n-1)d 2
Sn 302x3(30-1)6 2
Sn 156(29x6)
Sn 15x180 2700
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EXAMPLE 2a)Find the nth term of the arithmetic
series 71115..b)Which term of the sequence is
equal to 51?c)Hence find 7111551
a) a7, d4 so the nth term is 4n3
b) 4n3 51 4n 48 (subtract 3) n
12 (divide by 4)
c) Using the formula Sn n2a(n-1)d a7,
d4 and n12 2
Sn 122x7(12-1)4 2
Sn 614(11x4)
Sn 6x58 348