GIVEN:
We are given a Fibonacci sequence as shown in the attached image.
Required;
To use the pattern derived to find the sum of the squares of the first 16 Fibonacci numbers.
Step-by-step solution;
We have a Fibonacci sequence whose first term is 1.
The sequence and the sum of the squares of a given number of terms is derived as follows;
[tex]\begin{gathered} 1^2+1^2=1\times2 \\ \\ 1^2+1^2+2^2=2\times3 \\ \\ 1^2+1^2+2^2+3^2=3\times5 \\ \\ 1^2+1^2+2^2+3^2+5^2=5\times8 \\ \\ 1^2+1^2+2^2+3^2+5^2+8^2=8\times13 \\ \\ 1^2+1^2+2^2+3^2+5^2+8^2+13^2=13\times21 \end{gathered}[/tex]
Next, we determine the sequence from the 1st to 16th term as follows;
[tex]\begin{gathered} 1^2+1^2+2^2+3^2+5^2+8^2+13^2+21^2+34^2+55^2+89^2 \\ \\ +144^2+233^2+377^2+610^2+987^2=987\times1597 \end{gathered}[/tex]
The sum of the squares of the first 16 terms therefore is
[tex]987\times1597=1,576,239[/tex]
ANSWER:
[tex]1,576,239[/tex]
