please help me! finals tomorow :(

Question

find the first five terms of {2n/3n+1} a1=-2 an+3an-1?

Answer 1

Assuming the relationship is: 2n/(3n+1) a₁ = 3 a_n-1 - 2a_n; for n≥2, then

For n=2: 2x3 / (2x3 +1) a₁ = 3 a₁ -2 a₂; 6/7 a₁ = 3 a₁ -2 a₂; so a₂ = a₁ 17/14

For n=3 similarly we have a₃ = a₁ 213/140;

n=4 we have a₄ = a₁ 7187/1820;

n=5: a₅ = a₁ 19286 / 7280

All terms are calculated based on a₁; so we could choose a₁=1.

Therefore the first 5 terms are: {1, 17/14, 213/140, 7187/1820, 19286/7280}