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Number theory is a branch of pure mathematics devoted to the study of the properties and relationships of natural numbers. It's like the language of mathematics when it comes to understanding numbers themselves. Number theory explores things like divisibility, primes, and the solutions to equations in integers. In the context of the exercise, we particularly look at the sum of squares — a concept where we add up numbers of the form \( n^2 \). Understanding these sums is essential in number theory because they're related to various important theorems and principles, such as the Pythagorean theorem. When we sum squares, we're looking at how numbers interact and form patterns, which leads to deeper insights into how numbers work. Number theory forms the backbone of many mathematical proofs and conjectures, serving as an intellectual playground for mathematicians.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This is often referred to as the "common difference." Understanding this concept is crucial to solving the given exercise, particularly part (c), where the numbers 2, 5, 8, ..., 26 form an arithmetic sequence with a common difference of 3.

To visualize, if you start with the number 2 and always add 3, you'll continue through the sequence: 2, 5, 8, 11, etc., until you reach 26. Each term in this sequence can be calculated as \( a_n = a + (n-1)d \), where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Arithmetic sequences are used in various mathematical contexts, helping to simplify the calculation of sums, especially when these terms themselves are squared.

Natural numbers are the numbers that we use for counting, starting from 1 and moving upwards: 1, 2, 3, 4, etc. They are the foundational numbers used in mathematics. In this exercise, natural numbers are squared to test specific sums against given results.

The significance of natural numbers lies in their simplicity yet profound importance in number theory and arithmetic computations. When considered in sequences, they form the basis for understanding complex patterns and proofs, including the famous formula for the sum of the squares of the first \( n \) natural numbers: \[ S_n = \frac{n(n+1)(2n+1)}{6} \].This formula is especially useful for quickly calculating sums without directly adding each squared number, making it much easier to handle large numbers efficiently.

Mathematical proof is a logical argument that verifies the truth of a statement based on previously established statements, such as axioms and theorems. In mathematics, providing proof is essential to confirm that a particular statement or relation is universally valid. For the exercise, mathematical proof is the method by which we confirm that the sum of squares equals a given square.

Proofs can take many forms, including direct calculation, the use of established formulas, or even logical deduction. In our exercise, proofs are used to precisely check the relationships given, such as confirming that the sum of the sequence of squares matches a square number like \( 70^2 \) without directly summing each square. This can involve breaking down the big problem into smaller parts or using formulas effectively. Through proofs, students gain insights into the structure and beauty of mathematics.