In an ap, the sum of its first ten terms is -80 and the sum of its next ten terms is -280
find the ap.

The explicit formula for the AP is  -2n + 3.

1, -1 , -3, -5, -7, -9

Step-by-step explanation:

The sum of  the first n terms = n/2 [2a1 + (n - 1)d]   where a1=first term and d=common difference.

So n10 = 10/2( 2a1 + (10 - 1)d] = -80

n10 = 5(2a1 + 9d) = -80

The sum of 20 terms = -80 + -280 = -360.

So n20 = 20/2(2a1 + (20 - 1)d] = -360

n20 = 10(2a1 + 19d) = -360

Simplifying we have the systems

10a1 + 45d     = -80 (1)

20a1 + 190d  = -360(2)

Multiplying the first equation by 2:

20a1 + 90d   = -160(3)

Subtract (2) - (3):

100d = -200

d = -2.

and From equation (1):

10a1 =  (-80  -(45)*-2) =  10

so a1 = 1.

The explicit formula is 1 + (n - 1)-2

= -2n + 3.