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Proof by induction is a powerful mathematical technique used to prove that a statement is true for all natural numbers. It consists of two main steps.

The first step is the base case, where we test if the statement holds true when applied to the smallest natural number, often n=1. For our exercise, when n=1, both sides of the equation equaled 7, confirming that the base case is true.

In the inductive step, we assume the statement is true for some arbitrary natural number k, called the induction hypothesis. Then we prove that if it's true for k, it must be true for k+1. By adding the k+1 term to the assumed true equation, and simplifying, we can demonstrate that the statement holds.

If both the base case and the inductive step are proven, the statement is true for all natural numbers. This logical domino effect ensures that if it's true for 1, then for 2, then for 3, and so on, thus completing the proof by induction.

Arithmetic sequences are a list of numbers where each term is derived by adding a constant to the previous term. This constant is known as the common difference. In the given exercise, the sequence 7, 5, 3,... follows a pattern where the common difference is -2, as each term is decreased by 2 from the previous one.

The general form of an arithmetic sequence can be expressed as: \[a_n = a_1 + (n - 1) \times d\], where \(a_n\) is the nth term, \(a_1\) is the first term, and d is the common difference.

The sum of an arithmetic sequence can also be found using a formula involving the number of terms, the first term, and the last term. It's crucial in solving exercises like ours, where the sum of a sequence is equated to an algebraic expression.

Algebraic expressions are mathematical phrases that can contain numbers, operators, and variables. They don't have an equals sign as equations do. Examples include \(2x + 3\) or \(y^2 - 5y + 7\). In our context, the algebraic expression \(-n^2 + 8n\) represents the sum of the terms of the given sequence.

When working with algebraic expressions, it's important to understand how to simplify them. This involves combining like terms, applying the distributive property, and performing operations with the same order of precedence as in arithmetic.

The ability to manipulate algebraic expressions is vital when conducting a proof by induction, as it allows us to demonstrate that both the assumed hypothesis and the next logical step are equivalent, thereby showing the expression holds for all terms.