4.3.4 Sum of the Squares of Arbitrary Fibonacci Sequence

For the sum of the squares of the first $n$ terms of an Arbitrary Fibonacci Sequence:

The Formula

$$A_1^2 + A_2^2 + \dots + A_n^2 = A_n \times A_{n+1} - A_1(A_2 - A_1)$$

Special Case: Standard Fibonacci

For the Standard Fibonacci Sequence, $F_2 - F_1 = 0$, so the formula simplifies to:

$$F_1^2 + F_2^2 + \dots + F_n^2 = F_n \times F_{n+1}$$

Example: Standard Fibonacci

$$1^2 + 1^2 + 2^2 + 3^2 + \dots + 21^2 + 34^2 = 34 \times 55 = 1870$$

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Problem Set 4.3.4

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