Hi I am doing a research which involves matrices, and I wish to find the relation between $b$ and $c$. The nxn matrix is given as follows: $$ \begin{bmatrix} 1/2 & -c & 1/2 & 0 & 0 & 0 & ... & 0 \\ 0 & 1/2 & -c & 1/2 & 0 & 0 & ... & 0 \\ 0 & 0 & 1/2 & -c & 1/2 & 0 & ... & 0 \\ ...\\ 0 & 0 & 0 & 0 & ... & 0 & 1/2 & -c \\ 0 & 0 & 0 & 0 & 0 & ... & 0 & 1/2 \\ \end{bmatrix} \cdot \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \\ ...\\ b_{N-2} \\ b_{N-1} \\ \end{bmatrix} = \begin{bmatrix} a_{1} \\ a_{2} \\ a_{3} \\ ...\\ a_{N-1} \\ a_{N} \\ \end{bmatrix}$$
It is crucial for me to find the relation between $b$ and $c$. Can someone help me drop some hints on how I can achieve that. Sorry the matrix is not well written. It is suppose to be in the form of $A\tilde{b}=\tilde{a}$.
Believing Lord Shark's version, you can just do back substitution, starting at the bottom. You get $$\frac 12b_N=a_N\\b_N=2a_N\\\frac 12b_{N-1}-cb_N=a_{N-1}\\b_{N-1}=2a_{N-1}+2cb_N\\ \frac 12b_{N-2}-cb_{N-1}+\frac 12b_N=a_{N-2}\\ b_{N-2}=2(a_{N-2}+cb_{N-1})-b_N$$ and so on up the line