# Arithmetic progression. I have to find out the first term and the ratio! We know the third term = 8 and the seventh term of progression= 20.

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We know the formula for any term of an arithmetic progression:

an=a1 + (n-1)r, where a1 is the first term and r is the ratio.

a3=a1 + (3-1)r

a7=a1 + (7-1)r

Doing the substitution where we know the values

8 = a1 + 2r

20 = a1 + 6r

We are subtracting the second relation from the first one and the result will be

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

-12 = -4r

r=3

With the known value of r, we go back in the first relation and making substitution of r

8 = a1 + 2r

8 = a1 + 2x3

8 = a1 + 6

a1= 8-6

a1=2

Formula of arithmetic progression of the nth term would be:

an=a1+(n-1)s

where a1: the initial term of an arithmetic progression

s: common difference of successive numbers

You know the third term is 8 so:

8= a1+(3-1)s

8=a1+2s

You know that the seventh term is 20 so:

20= a1+(7-1)s - eqn 1

20=a1+6s - eqn 2

Subtract eqn 1 from eqn 2

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

12=4s

4s=12

s=3//

So, use this value and substitute into eqn 2

20= a1+6(3)

a1+18=20

a1=20-18

a1=2//

The first term would be** 2**