Descending arithmetic sequence second and third term multiplication is 21. The sum of the first seven terms is -7. Find the 5th term of this sequence.

Answer 1

an = a1 + d(n-1)

a5 = a1 +d(5-1)

a5 = a1 +4d

sn = (n/2)(a1 + an)

s7 = (7/2)(a1 +a1+d(6))=-7

7(2a1 +6d) =-14

2a1 +6d = -2

a1 +3d =-1

a1 =-1-3d

a5 =1-3d +5d

a5 =1+2d

a2(a3) = 21

(-1-3d +d)(-1-3d+2d) = 21

(-1-2d)(-1-d) =21

1+3d +2d^2 =21

2d^2 +3d -20 =0

(2d-5)(d+4) =0

d =-4 = common difference

a1= -1-3(-4) = 11 = 1st term

a5 = 11+4(-4) =-5 = the 5th term

11, 7, 3, -1, -5, -9, -13 all sum to -7

a2 =7

a3 =3

a2a3 =(7)(3)=21