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I didn't "solve" for it; I just rewrote the second expression in a different way. You should convince yourself that the two expressions are equal.odolwa99 said:
Proof by induction is a mathematical technique used to prove that a statement is true for all natural numbers. It involves breaking down a problem into smaller cases and using a logical argument to show that the statement holds for each case, ultimately proving that it holds for all natural numbers.
Proof by induction works by first proving that the statement is true for the base case, typically when n = 1. Then, assuming that the statement is true for some arbitrary value of n, the goal is to prove that it is also true for n+1. By showing that the statement holds for both the base case and for n+1, it can be concluded that the statement is true for all natural numbers.
Some common types of divisibility proofs using induction include proving that a statement is true for all even or odd numbers, or for all multiples of a certain number. These types of proofs often involve using the fact that if a number is divisible by another number, then its multiples are also divisible by that number.
The steps to proving a divisibility statement using induction are as follows:
1. Prove that the statement is true for the base case (usually when n = 1).
2. Assume that the statement is true for some arbitrary value of n.
3. Use this assumption to prove that the statement is also true for n+1.
4. Conclude that the statement is true for all natural numbers by the principle of mathematical induction.
Proof by induction is a powerful tool in mathematics because it allows us to prove that a statement is true for infinitely many cases (all natural numbers) by only considering a few specific cases. This saves time and effort compared to proving the statement for each individual case. Additionally, many mathematical theorems and properties can be proven using induction, making it a widely applicable technique.
