A sequence is defined by the recurrence relation $u_{n+1} = 0.2u_n + 9$, ${u_5 = 11}$. What is the value of ${u_3}$?
I have not encountered a problem like this when only one value for n is provided. Normally I am given a couple of values for n and I can solve for u by solving simultaneous equations.
I thought I could put 11 into the equation and substitute 5 for n and find u.
${11 = 0.2u(5) + 9}$
=> ${u = 2}$
${u_3 = 0.2(3)(2) + 9} = 10.2$
But the answer is incorrect.
Note that I have edited this answer since OP posted an image of the original question.
According to the recurrence relation, we have $u_5 = 0.2u_4 + 9$ and $u_4 = 0.2u_3 + 9$. Then we have $$\begin{align} u_5 &= 0.2u_4 + 9 \\ u_5 &= 0.2(0.2u_3 + 9) + 9 \\ (11) &= 0.2(0.2u_3 + 9) + 9 \\ 2 &= 0.2(0.2u_3 + 9) \\ 2/(0.2) &= (0.2u_3 + 9) \\ 10 &= 0.2u_3 + 9 \\ 1 &= 0.2u_3 \\ 1/(0.2) &= u_3 \\\\ 5 &= u_3 \end{align}$$