Lesson BASICS - Arithmetic Series

Source code of 'BASICS - Arithmetic Series'

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Introduction
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.

Definitions
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

nth Term
Algebraically:
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

--> {{{ nth term = a + (n-1)d }}}

EXAMPLES
Q. Find the 12th term of the sequence 1,5,9,13,...

A. a=1, d=4, n=12.
{{{ 12th term = 1 + (12-1)*4 }}}
{{{ 12th term = 1 + (11)*4 }}}
{{{ 12th term = 1 + 44 }}}
{{{ 12th term = 45 }}}

which we can verify by writing out the first 12 terms in full:
1,5,9,13,17,21,25,29,33,37,41,45

Summation
The summation of n terms is given by:

{{{ Sn = (n/2)(2a + (n-1)d) }}}

which is also written as {{{ Sn = (n/2)(a + L) }}} where L is the last term, defined as L=a+(n-1)d

EXAMPLES
Q. Find the sum of the first 10 terms of the sequence 1,5,9,13,17,...

a=1, d=4, n=10
--> {{{ Sn = (n/2)(2a + (n-1)d) }}}
{{{ Sn = (10/2)(2(1) + (10-1)4) }}}
{{{ Sn = (5)(2 + (9)4) }}}
{{{ Sn = (5)(2 + 36) }}}
{{{ Sn = (5)(38) }}}
Sn = 190

Further Example
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 = (8/2)(2(2) + (8-1)3) }}}
{{{ Sn = (4)(4 + (7)3) }}}
{{{ Sn = (4)(4 + 21) }}}
{{{ Sn = (4)(25) }}}
Sn = 100

Summary
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.