Correct Application of Inductive Hypothesis

Question

To provide full context of the practice question I'm attempting, it is as follows:

For every integer n such that n ≥ 8, there exist nonnegative integers a_n and b_n such that 3a_n + 5b_n = n.

Write a proof of the claim using the strong form of mathematical induction with the integer 10 as breakpoint — such that that n = 8, n = 9 and n = 10 would all be considered in the basis.

This is my attempt at the proof, but I can't seem to get past applying the I.H. in order to reach my goal in the inductive step.

Basis
If n = 8, then 3a₈ + 5b₈ = 8
If n = 9, then 3a₉ + 5b₉ = 9
If n = 10, then 3a₁₀ + 5b₁₀ = 10

Inductive Hypothesis
Let k be an integer such that k ≥ 10. It is necessary and sufficient to use the following:

• Inductive Hypothesis: 3a_n + 5b_n = n for every integer n such that 8 ≤ n ≤ k.

Inductive Claim
3a_{k+1 + 5bk+1} = k+1

Now I have to complete the remainder of the proof but I'm not even sure if I was proving it correctly so far or how to complete it from where I'm at.

Any guidance would be much appreciated!

Answer 1

Note that, in particular, there are $a_{k-2}$ and $b_{k-2}$ such that $3a_{k-2} + 5b_{k-2} = k-2$. Now add $3$ to both sides.

Answer 2

For the step, see answer by @user3482749

But other than the inductive step, you also need to show that you can find actual values for $a_n$ and $b_n$ (and the others) in your base cases to make sure these work!

For example, to satisfy that $3a_8+5b_8=8$, you can set $a_8=1$ and $b_8=1$

Now do this for the other base cases as well