Starting Fibonacci
Problem
The standard Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ... begins with two 1s and each later number in the sequence is the sum of the previous two numbers.
Other Fibonacci-like sequences can be constructed by starting with any two numbers a and b, and using the same rule for creating the other numbers in the sequence.
What is the first term of the Fibonacci-like sequence whose second term is 4 and whose fifth term is 22?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 5
$\underline{\quad\quad\ }\quad\underline{\ \ \ \ 4\ \ \ \ }\quad\underline{\quad\quad\ }\quad\underline{\quad\quad\ }\quad\underline{\ \ \ 22\ \ \ }$
$\underline{\ \ \ x\ \ \ }\quad\underline{\ \ \ \ 4\ \ \ \ }\quad\underline{x+4}\quad\underline{x+8}\quad\underline{\ \ \ 22\ \ \ }$
$x+4\ \ +\ \ x+8\ \ =22\\
\Rightarrow 2x+12=22\\
\Rightarrow 2x=10\\
\Rightarrow x=5$