Is this a correct recursive sequence definition?

Question

Take this definition:

Is this definition of $s_k$ for $k\ge2$ correct?

$s_k=6a_{k-1}-5a_{k-2}$ but where does the $a$ term come from?

The book swaps $a$ and $s$ interchangeably.

Answer 1

If the $(a_k)_k$ sequence is well-defined there is nothing wrong with that definition. You should give us more details on $(a_k)_k$ if you want some more help.

If $a$ were to be replaced by $s$ in the definition, it would still be correct.

Answer 2

If $a$ is a typo for $s$, as the comments say, this can be solved. Define the generating function $S(z) = \sum_{n \ge 0} s_n z^n$. Reorganize the recurrence so there are no subtractions in indices: $$ s_{n + 2} = 6 s_{n + 1} - 5 s_n $$ Multiply by $z^n$, sum over $n \ge 0$, recognize: \begin{align} \sum_{n \ge 0} s_{n + 1} z^n &= \frac{S(z) - s_0}{z} \\ \sum_{n \ge 0} s_{n + 2} z^n &= \frac{S(z) - s_0 -s_1 z}{z^2} \end{align} to end up with: $$ \frac{S(z) - 4 z}{z^2} = 6 \frac{S(z)}{z} - 5 S(z) $$ Solving for $S(z)$, as partial fractions: $$ S(z) = \frac{1}{1 - 5 z} - \frac{1}{1 - z} $$ This is just two geometric series: $$ s(n) = 5^n - 1 $$

Answer 3

Since it's been concluded that there was a typo, I'll offer an eigenvalue solution as an alternative to a generating function. So we have $\lambda^{2} - 6\lambda + 5 = 0$. This gives us eigenvalues of $\lambda = 1, 5$.

We then set up our general form equation: $s_{k} = A + B * 5^{k}$.

Solving $s_{0} = 0 = A + B$, we get $A = -B$.

Then we solve $s_{1} = -B + 5B = 4$, so $B = 1$.

And so $s_{k} = 5^{k} - 1$, the same solution given by the generating function.