ask expert: be as specific as possible: Mathematical Induction

a$)$ Prove the following statement using mathematical induction: For all positive integers n$,$ the sum of the squares of the first n natural numbers is given by the formula: n$($n $+1)(2$n $+1)/6.$

b$)$ Explain the principle of mathematical induction. Why is it particularly useful in the field of Computer Science?

c$)$ Consider a network of computers where each computer can send a message only to its immediate neighbor. Design an algorithm to ensure a message sent from one end of the network reaches the other end. How can the principle of mathematical induction be used to verify that your algorithm correctly handles networks of increasing size?