the sum of the first twelve terms of arithmetic progression [AP] is 168. if the third term is 7. find the value of the common difference and the first term

Common difference = 2

First term = 3.

Step-by-step explanation:

3rd term = a1 + (3 - 1) d = 7 where a1 = first term and d = the common difference.

Sum of the first 12 terms = (12/2) [2a1 + (12-1) d] = 168

so 6 (2a1 + 11d) = 168

So simplifying, we have the following system:

a1 + 2d = 7 ... (A)

12a1 + 66d = 168

2a1 + 11d = 28 ... (B) Multiplying equation A by - 2:

-2a1 - 4d = - - 14 Adding this to equation B

7d = 14

d = 2

Now plug this into equation A:

a1 + 2 (2) = 7

a1 = 7-4

a1 = 3.