Question

If y₁ < y₂ are arbitrary real numbers and yₙ = (1/3 yₙ-₁) + (2/3 yₙ-₂) for n > 2,

Find the limit of (yₙ).

Answer 1

The limit of the sequence (
`y_n`) as (n) approaches infinity is zero, as determined by the characteristic equation and initial conditions. The term
`\(\left((2)/(3)\right)^n\)` tends to zero, ensuring the overall limit is zero.

To find the limit of the sequence (
`y_n`), let's first express the recursive relation in terms of the characteristic equation. The given recursive relation is:

`\[ y_n = (1)/(3) y_(n-1) + (2)/(3) y_(n-2) \]`

To find the characteristic equation, assume a solution of the form
`\(y_n = r^n\)`. Substitute this into the equation:

`\[ r^n = (1)/(3) r^(n-1) + (2)/(3) r^(n-2) \]`

Multiply through by
`\(3r^(n-2)\)` to get rid of the fractions:

`\[ 3r^n = r^(n-1) + 2r^(n-2) \]`

Now, rearrange the terms:

`\[ 3r^n - r^(n-1) - 2r^(n-2) = 0 \]`

This is the characteristic equation. Factor it:

`\[ r^(n-2)(3r^2 - r - 2) = 0 \]`

Now solve for the roots (r). You can factor the quadratic or use the quadratic formula:

`\[ r = (1 \pm √(1 + 4(3)(2)))/(6) \]`

You'll find two distinct roots for (r), and let's call them
`r_1` and
`\(r_2\)`.

`\[ r_1 = (-1 + √(25))/(6) = (2)/(3) \]`

`\[ r_2 = (-1 - √(25))/(6) = -1 \]`

The general solution for
`\(y_n\)` is then:

`\[ y_n = A \left((2)/(3)\right)^n + B(-1)^n \]`

Now, we need to find the specific values of (A) and (B) based on the initial conditions
`\(y_1\)` and
`\(y_2\)`.

Given that
`\(y_1 < y_2\)`, we can write the following:

`\[ y_1 = A\left((2)/(3)\right) + B(-1) \]`

`\[ y_2 = A\left((2)/(3)\right)^2 + B(-1)^2 \]`

Solve these two equations for (A) and (B), and substitute them back into the general solution.

After finding the values of (A) and (B), the limit as (n) approaches infinity will depend on the values of the roots
`\(r_1\)` and
`\(r_2\)`. Since
`\(r_1 = (2)/(3)\)` and
`\(|r_2| < 1\)`, the term
`\(B(-1)^n\)` will tend to zero as (n) approaches infinity.

Thus, the limit of
`\(y_n\)` as (n) approaches infinity is given by the term
`\(A\left((2)/(3)\right)^n\)`. Since
`\(\left|(2)/(3)\right| < 1\)`, this term will approach zero, and the limit of
`\(y_n\)` will be zero.