A sequence an is defined recursively by a1 2 a2 3 and an

A sequence {a_n} is defined recursively by a_1 = 2, a_2 = 3 and a_n = 3a_n-1 -2a_n-2 for n > 3. Conjecture a formula for a_n and verify that your conjecture is correct.

Solution

a₁ = 2

a₂ = 3

a_n = 3a_n-1 – 2a_n-2

a₃ = 3a₂ – 2a₁

a₃ = 3*3 – 2*2 = 5

a₄ = 3*5 – 2*3 = 9

a₅ = 3*9 – 2*5 = 17

a₆ = 3*17 – 2*9 = 33

a₇ = 3*33 – 2*17 = 65

a₈ = 3*65 – 2*33 = 129

The formula for n >= 3 is

a_n = 2a_n-1 – 1

a₃ = 2a₂ – 1

a₃ = 2*3 – 1 = 5

a₄ = 2*5 -1 = 9

a₅ = 2*9 – 1 = 17

a₆ = 2*17 – 1 = 33

a₇ = 2*33 – 1 = 65

a₈ = 2*65 – 1 = 129

So the formula conjectured for n >= 3, a_n = 2a_n-1 – 1 is correct.