The 7 th term of an AP is four times it's 2nd term is two more than three times of it's 4th term .Find the progression.​

The progression is :2,5,8,11,14,17

The progression.

Tn = nth term

a = First term

n = Number of terms

d = Common difference

The seventh term of an AP is four times its 2nd term.

⟹T₇ = 4 T₂

⟹a + (7 - 1)d = 4{a+(2 -1)d}

⟹a + 6d = 4{a+d}

⟹4a - a = 6d - 4d

⟹3a = 2d_______(i)

12th term is 2 more than three times of its 4th term.

⟹T₁₂ = 3T₄ + 2

⟹a + (12 - 1) d =3{a + (4 - 1)d}+2

⟹a+11d = 3{a +3d}+2

⟹a + 11 d = 3a + 9d + 2

⟹11d - 9d = 3a - a + 2

⟹2d = 2a + 2⠀⠀⠀⠀ [From eqⁿ (i)2d = 3a]

⟹3a = 2a + 2

⟹3a - 2a = 2

⟹a = 2

⠀⠀⠀⠀First term= 2

In equation (i),

⟹2d = 3a

⟹2d = 3×2⠀⠀⠀⠀⠀ [ The value of a = 2]

⟹2d = 6

⟹d = 3

⠀⠀⠀⠀Common difference = 3

⟹T₁ = a = 2

⟹T₂ = a + d = 2 + 3 = 5

⟹T₃ = a + 2d = 2 + 6 = 8

⟹T₄ = a + 3d = 2 + 9 = 11

⟹T₅ = a + 4d = 2 + 12 = 14

⟹T₆ = a + 5d = 2 + 15 = 17