A series of natural numbers F1, F2, F3, F4, F5, F6, F7, ... obeys F(n+1) = F(n) + F(n-1) for all integers n ≥ 2. If F6 = 37, and F7 = 60, then what is F1?

Steps to Solve

1. Write down the recursive relation

The recursive relation is given as:

$$ F(n+1) = F(n) + F(n-1) $$

This means that each term in the series is the sum of the two preceding terms.

2. Identify known values

From the problem, we know:

• $F_6 = 37$
• $F_7 = 60$

3. Find $F_5$ using $F_6$ and $F_7$

Using the recursive relation for $n = 6$, we can find $F_5$:

$$ F_7 = F_6 + F_5 $$

Substituting in the known values:

$$ 60 = 37 + F_5 $$

To isolate $F_5$, rearrange the equation:

$$ F_5 = 60 - 37 $$

Calculating this gives:

$$ F_5 = 23 $$

4. Find $F_4$ using $F_5$ and $F_6$

Next, we find $F_4$ using $F_5$ and $F_6$ for $n = 5$:

$$ F_6 = F_5 + F_4 $$

Substituting the known value of $F_5$:

$$ 37 = 23 + F_4 $$

Rearranging gives:

$$ F_4 = 37 - 23 $$

Calculating this gives:

$$ F_4 = 14 $$

5. Find $F_3$ using $F_4$ and $F_5$

Now, for $n = 4$, we find $F_3$:

$$ F_5 = F_4 + F_3 $$

Substituting the known value of $F_4$:

$$ 23 = 14 + F_3 $$

Rearranging gives:

$$ F_3 = 23 - 14 $$

Calculating this gives:

$$ F_3 = 9 $$

6. Find $F_2$ using $F_3$ and $F_4$

Next, for $n = 3$, we find $F_2$:

$$ F_4 = F_3 + F_2 $$

Substituting the known value of $F_3$:

$$ 14 = 9 + F_2 $$

Rearranging gives:

$$ F_2 = 14 - 9 $$

Calculating this gives:

$$ F_2 = 5 $$

7. Find $F_1$ using $F_2$ and $F_3$

Finally, for $n = 2$, we find $F_1$:

$$ F_3 = F_2 + F_1 $$

Substituting the known value of $F_2$:

$$ 9 = 5 + F_1 $$

Rearranging gives:

$$ F_1 = 9 - 5 $$

Calculating this gives:

$$ F_1 = 4 $$