# What are 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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The question is basically asking:

Term no: 1, 2, 3, 4, 5, 6, 7, ...

Nos: ?, _, 8, _, _, _, 20, ...

By inspection,

Common diff

=(20-8)/(7-3)

= 12 / 4

= 3

1st term

= 8 - 2x3

= 8 - 6

= 2

The 3rd term of an AP is 8 and the 7th term of the AP is 20. Required to find the 1st term and the common ratio.

The nth term of an AP is given by an = a1+(n-1) d, where a1 is the 1st term and d is the common ratio.

Therefore a3 = a1 + (3-1)d = 8...(1)

a7 = a1+(7-1)d = 20...(2)

(2)- (1) gives: (6-2) d = 20-8 = 12.

Therefore d = 12/(6-2) = 3.

Therefore from (1), we get a1+(3-1)d = 8. So a1 = 8- (3-1)d = 8- 2*3 = 2.

Therefore a1 = 2 and d = 3.

Therfore the first term of the AP is a1 = 2. The common ratio of the AP is d= 3.

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.**