Arithmetic Series or Progressions are sequences of numbers that increment by a fixed common difference eg 4,7,10,13,16,... is an Arithmetic series with common difference 3.
Let d = common difference
Let a = first term
Let n = number of terms
Let Sn = sum of first n terms
So, in the sequence -2,3,8,13,18,... we have:
first term is -2 --> a = -2
difference is 5 --> d = 5
1st term = a
2nd term = a + d
3rd term = a + 2d
4th term = a + 3d
5th term = a + 4d
6th term = a + 5d
and so on.
In general, the nth term is given as a + (n-1)d
Q. Find the 12th term of the sequence 1,5,9,13,...
A. a=1, d=4, n=12.
which we can verify by writing out the first 12 terms in full:
The summation of n terms is given by:
which is also written as
where L is the last term, defined as L=a+(n-1)d
Q. Find the sum of the first 10 terms of the sequence 1,5,9,13,17,...
a=1, d=4, n=10
Sn = 190
Q. The second term of an arithmetic series is 5 and the fifth term is 14. Find the common difference, the tenth term and the sum of the first 8 terms.
We do not know a, which is crucial in all the calculations, so we are aiming to find that as well as d.
So, what we do know is the following:
2nd term, a+d = 5
5th term, a+4d = 14
subtract these to give 3d = 9 --> d = 3
So, using this in a+d=5 we have
a+3 = 5
--> a = 2
The 10th term = a+9d
The 10th term = 2+9(3)
The 10th term = 2+27
The 10th term = 29
Now for the sum of 8 terms:
Sn = 100
This is the introduction to Arithmetic Series'. However, with the question quoted at the end here, there is not a lot of things that you need to know. It is all about practice now: practice using the "nth term" and "summation" formulae.
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