# A.P.Determine the first term and the common difference of an A.P. if the third term is 8 and seventh term is 20

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You need to use the equation that relates the given members of the arithmetical series, such that:

`a_7 = a_3 + 4d`

The problem provides the members `a_7` and `a_3` , such that:

`20 = 8 + 4d => 4d = 20 - 8 => 4d = 12 => d = 3`

You need to use the equation that relates the members `a_3` and `a_1` , such that:

`a_3 = a_1 + 2d => 8 = a_1 + 2*3 => a_1 = 8 - 6 => a_1 = 2`

**Hence, evaluating the member `a_1` and common difference `d` yields `d = 3` and **`a_1 = 2.`

We'll write the formula for the general term of an arithmetic progression:

an=a1 + (n-1)d, where a1 is the first term and d is the common difference.

a3=a1 + (3-1)d

a7=a1 + (7-1)d

We'll substitute a3 and a7 by the values given in enunciation:

8 = a1 + 2d

20 = a1 + 6d

We'll subtract the second relation from the first one and we'll get:

8-20 = a1 + 2d -a1 - 6d

We'll eliminate and combine like terms:

-12 = -4d

d=3

We'll substitute d in the first relation:

8 = a1 + 2d

8 = a1 + 2*3

8 = a1 + 6

a1= 8-6

a1=2

The first term and the common difference of the a.p. are: a1 = 2 and d = 3.