Sequence of numbers recurrence relation

Question

A sequence of real numbers $$ u_1, u_2, u_3... $$ satisfies $$u_1=1$$ and the recurrence relation $$4u_{n+1}=au_n-2$$ for all positive integers n where a is a real constant. Express $$u_n$$ in terrms of a and n

Answer 1

Hint: This is a first order linear recurrence relation. It's solved as first order differential equations with constant coefficients:

• First solve the linear recurrence without constant term: $u_{n+1}=\dfrac a4\,u_n$,
• Look for one particular solution of the complete linear recurrence. This should be a constant if $a\neq 4$,
• Examine the particular case $a=4$.

Answer 2

Hint: Try writing out the first few terms of the sequence, in terms of $a$.