# Math

The sixth term of an arithmetic sequence is 20.6 and the 9th term is 30.2. Find the 20th term and find the nth term.

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1. In arithmetic sequence :

an = a1 + ( n - 1 ) * d

d is the common difference

n is the number of the term to find

a6 = a1 + ( 6 - 1 ) * d = 20.6

a1 + 5 d = 20.6

a9 = a1 + ( 9 - 1 ) * d = 30.2

a1 + 8 d = 30.2

Now you must solve system of two equations with two unknown :

a1 + 5 d = 20.6

a1 + 8 d = 30.2

The solutions are :

a1 = 23 / 5 = 4.6

d = 16 / 5 = 3.2

a20 = a1 + ( n - 1 ) * d

a20 = 4.6 + ( 20 - 1 ) * 3.2

a20 = 4.6 + 19 * 3.2

a20 = 4.6 + 60.8

a20 = 65.4

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2. an = a1 + ( n - 1 ) * d

an = 4.6 + ( n - 1 ) * 3.2

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3. In an arithmetic sequence,
an = a1 + (n-1)d

where
an = nth term
a1 = first term
n = number of terms
d = common difference

The sixth term is 20.6 so,
20.6 = a1 + (6-1)d
20.6 = a1 + 5d

The ninth term is 30.2 so,
30.2 = a1 + (9-1)d
30.2 = a1 + 8d

Now you have two equations, to unknowns. We can subtract them to solve for d:
20.6 = a1 + 5d
-(30.2 = a1 + 8d)
--------------------------
-9.6 = -3d
d = 3.2

And thus,
20.6 = a1 + 5(3.2)
a1 = 4.6

Now you have values for a1 and d, you can solve for the 20th term.

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