Find the common difference and the nth       term of each arithmetic sequence     a. 3,5,7,9     b. 5,2, -1,-4?

a.
Common difference, d
= 5 - 3
= 2

First term, a = 3

nth term
= a + (n - 1) d
= 3 + (n - 1) × 2
= 3 + 2n - 2
= 2n + 1

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b.
Common difference, d
= 2 - 5
= -3

First term, a = 5

nth term
= a + (n - 1) d
= 5 + (n - 1) × (-3)
= 5 - 3n + 3
= 8 - 3n

An arithmetic sequence is one where you add the same amount to get to the next term.

Part a:

In the first sequence you are adding 2 each time, so the common difference is 2.
d = 2

And the first term is 3:
a = 3

The general formula for the nth term of an arithmetic sequence is:
a[n] = a + d(n - 1)

Plug in your numbers:
a[n] = 3 + 2(n - 1)

If you want, you can simplify that slightly:
a[n] = 3 + 2n - 2
a[n] = 2n + 1

Double-check:First term --> a[1] = 2(1) + 1 = 3
Second term --> a[2] = 2(2) + 1 = 5
etc.

Part b:

Here the common difference is negative. Can you figure it out? If you can't take any term and subtract the one before it.
2 - 5 = ?
or
-1 - 2 = ?

And you can obviously figure out the first term just by looking.

Now plug that into the nth term formula.

You should practice this, so try the second one on your own.

a.  AP is 3,5,7,9,.....Common difference between consecutive terms = d, say, = 2. 1st term is a, say, = 3. nth term is a +(n-1)d = 3+(n-1)2 = 2n +1, n = 1,2,3,....

b. AP is 5,2,-1,-4,....Common difference between consecutive terms = d, say, = -3. 1st term is a, say, = 5. nth term is a +(n-1)d = 5+(n-1)(-3) = 8-3n, n = 1,2,3,...

a. 
Common difference is '2' 
nth term is 2n + 1 

b.
Common difference is '-3' 
nth term is -3n + 8

Method ; 
Note the difference between each term.  The answer is the coefficient of 'n' 
So we write 
(difference)n + c = first value.
Hence 'c' (the constant) is first term - (difference)n 

Hope that helps!!!!! 

  a. 
  3, 5, 7, 9, ... 
  a_n = 2 n + 1 
  The difference between all of the adjacent terms
  is the same and equal to
  d = 2  b.   5, 2, -1, -4, ...
  a_n = 8 - 3 n 
  d = -3

a) nth term of AP is a + (n - 1)d

so, 3, 5, 7, 9,...a = 3 and d = 2

Hence, 3 + 2(n - 1) => 2n + 1

b) with 5, 2, -1, -4,....a = 5 and d = -3

Hence, 5 + (n - 1)(-3) => 8 - 3n

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