Solve the recurrence an 2 a2n a3n 1 with a1 1 and a2 2S

Solve the recurrence an + 2 a2n = a3n + 1 with a1 = 1 and a2 = 2.

Solution

Given that a_n+2a_n2 = a³_n+1

Take log on both sides

ln an+2 + 2ln an = 3 ln an+1

Or ln an+2 = -2ln an + 3ln an+2

Let n =1

ln a3 = -2 ln 1+3ln 2 = 3ln 2

a3 =8

lna4 = -2ln2+3ln 8

a4 = 128

ln a5 = -2ln8+3ln 128

a5 = 128^3/64 = 32768 and so on.

In general

ln an+2 = 3 ln an+1-2ln an

= 3(3ln an-2ln an-1) -2ln an

and so on.