2.2.2 Fibonacci Numbers

It would be best to have the Fibonacci numbers memorized up to $F_{15}$ because they crop up every now and then on the number sense test. In case you are unaware, the fibonacci sequence follows the recursive relationship of $F_n = F_{n-1} + F_{n-2}$. The following is a helpful table:

n	$F_n$	n	$F_n$	n	$F_n$
1	1	6	8	11	89
2	1	7	13	12	144
3	2	8	21	13	233
4	3	9	34	14	377
5	5	10	55	15	610

The most helpful formula to memorize concerning Fibonacci Numbers is the the sum of the first n Fibonacci Numbers is equal to $F_{n+2} - 1$.

Sum of General Fibonacci Sequence

A common problem asked on the latter parts of the number sense test is: Find the sum of the first eight terms of the Fibonacci sequence 2, 5, 7, 12, 19, …

Method 1: General Formula

The sum of a the first n-terms of a general Fibonacci sequence $a, b, a+b, a+2b, 2a+3b, \dots$ is: $\text{Sum} = a \cdot (F*{n+2} - 1) + d \cdot (F*{n+1} - 1)$ Where $d = (b - a)$.

So for our example: $\text{Sum} = 2 \cdot (F\_{10} - 1) + (5 - 2) \cdot (F_9 - 1) = 2 \cdot 54 + 3 \cdot 33 = 108 + 99 = 207$

Method 2: Specific Formulas

The following is a list of the sums of a general Fibonacci sequence for 1-12 terms:

n	Sum Formula
1	$F_1 \cdot a$
2	$F_3 \cdot a + F_2 \cdot b$
3	$2 \cdot F_3$
4	$4 \cdot F_3 - a$
5	$7 \cdot F_3 - 2a$
6	$4 \cdot F_5$
7	$4 \cdot F_6 + a$
8	$7 \cdot F_6 - 2b$
9	$7 \cdot F_7 - F_4$
10	$11 \cdot F_7$
11	$11 \cdot F_8 + a$
12	$18 \cdot F_8 - b$

So in our case, we are summing the first 8 terms, which is just $7 \cdot F_6 - 2b$, where $F_6$ represents the sixth term in the sequence of 2, 5, 7, 12, 19, … (which is 31), so $7 \cdot 31 - 2 \cdot 5 = 217 - 10 = 207$.

Problem Set 2.2.2