Find the first four terms of the sequence defined by a1 = 3 and an = 3

1.	Find the first four terms of the sequence defined by a1 = 3 and an = 3an–1 + 2, for all n > 1.
Answer» Given, a₁ = 3 and a_n = 3a_n–1 + 2, for all n > 1 We can find the first four terms of a sequence by putting values of n from 1 to 4 When n = 1 : a₁ = 3 When n = 2 : a₂ = 3a_2–1 + 2 ⇒ a₂ = 3a₁+ 2 ⇒ a₂ = 3(3) + 2 ⇒ a₂ = 9 + 2 ⇒ a₂ = 11 When n = 3 : a₃ = 3a_3–1 + 2 ⇒ a₃ = 3a₂+ 2 ⇒ a₃ = 3(11) + 2 ⇒ a₃ = 33 + 2 ⇒ a₃ = 35 When n = 4 : a₄ = 3a_4–1 + 2 ⇒ a₄ = 3a_{3 + 2} ⇒ a₄ = 3(35) + 2 ⇒ a₄ = 105 + 2 ⇒ a₄ = 107 ∴ First four terms of sequence are 3, 11, 35, 107.

Find the first four terms of the sequence defined by a1 = 3 and an = 3an–1 + 2, for all n > 1.

Answer»

Given,

a₁ = 3 and a_n = 3a_n–1 + 2, for all n > 1

We can find the first four terms of a sequence by putting values of n from 1 to 4

When n = 1 :

a₁ = 3

When n = 2 :

a₂ = 3a_2–1 + 2

⇒ a₂ = 3a₁+ 2

⇒ a₂ = 3(3) + 2

⇒ a₂ = 9 + 2

⇒ a₂ = 11

When n = 3 :

a₃ = 3a_3–1 + 2

⇒ a₃ = 3a₂+ 2

⇒ a₃ = 3(11) + 2

⇒ a₃ = 33 + 2

⇒ a₃ = 35

When n = 4 :

a₄ = 3a_4–1 + 2

⇒ a₄ = 3a_{3 + 2}

⇒ a₄ = 3(35) + 2

⇒ a₄ = 105 + 2

⇒ a₄ = 107

∴ First four terms of sequence are 3, 11, 35, 107.