Question

Find the 4 number in ap such that sum of 2 and 3 term is 22 and the product of 1 and 4 term is 85

Answer 1

in AP

the sum of a2 and A3 is 22

so, a + b + a + 2D =2 2

a+ 3D=22 - a ....... first equation

now,

a× 4a=85

a×{a+3d}=85
now from equation 1 putting the value of a + 3D

22 a -a2 = 85

a2-22a+85=o
now factorise the equation. we have answer (a - 5 )or (a - 7 )then we observe,(a=7 or 5)

now, putting the value of a on the first equations.a=5
then we find the {d=4) .
so ,AP is 5,9,13,17.
I hope this answer is clear to you

Answer 2

ATQ:

a₂ + a₃ = 22

a + d  + a + 2d = 22

2a + 3d = 22--------------------------------(1)

And,

(a+d)(a+3d) = 85

a(a+3d) = 85

a(22-a) = 85                    (from 1)

22a - a² = 85

a²-22a+85

a-17a-5a+85

a(a-17)-5(a-17)

(a-5)(a-17)

a = 5      or a = 17

If a= 5

then,

2a+3d = 22

2*5 + 3d = 22

3d = 22-10

d = 12/3

d = 4

AP = 5,9,13,17........