Verify that the given equation holds for n=1, n=2, and n=3 ;...

Verify that the given equation holds for $n=1, n=2,$ and $n=3 ;$ and (b) use mathematical induction to show that the equation holds for all natural numbers.
$$F_{1}^{p}+F_{2}^{2}+F_{3}^{2}+\dots+F_{n}^{p}=F_{n} F_{n+1}$$

Key Concepts

Fibonacci Sequence

This is a well-known sequence where each term after the first two is the sum of the two preceding ones. It is often defined recursively with initial terms that are explicitly stated. The Fibonacci properties are useful in various mathematical identities, especially those involving sums of squares or other functions of the Fibonacci numbers.

Mathematical Induction

This is a proof technique that is used to establish the truth of a statement for all natural numbers. It involves two steps: first, showing that the statement holds for an initial value (the base case), and then proving that if the statement holds for an arbitrary natural number, it must also hold for the next number (the inductive step).

Base Case Verification

In the context of a proof by induction, the base case is the step where the proposition is confirmed for the first natural number (or other starting value). Establishing the base case is essential because it serves as the foundation on which the inductive argument is built.

Inductive Step

Once the base case is established, the inductive step involves assuming the statement holds for some arbitrary natural number (the induction hypothesis) and then proving it holds for the next number. This step shows that the truth of the statement perpetuates from one number to the next, thereby covering all natural numbers.

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