Find the twentieth and the nth of the sequence 4,9,14,19,....

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4,9,14,19,....a20,...an

We notice that the differenc ebetween terms is 5

==> r= 5

a1= 4

a2= 4+ 5 = 9

a3= 4+ 2(5) = 14

......

a20 = 4 + 19(5) = 99

an = a1 + (n-1)*r

      = 4 + (n-1)*5

      = 4+ 5n - 5

      = 5n -1

==> an = 5n -1

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First we have to find the difference between terms:

an = a1 + (n-1)d
d = (an - a1) / (n -1)
= (9 - 4) / 1
= 5

To test that this is an arithmetic series we can determine whether the differences are the same for the other two terms:

d = (14 - 4) / 2 = 5
d = (19 - 4) / 3 = 5

So it is an arithmetic series.  From here solving for a20 is as simple as using the same expression:

a20 = a1 + (20 - 1)d
a20 = 4 + 19(5)
a20 = 99

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