Chapter 8 of the F.Sc 1st Year Math curriculum delves into two essential topics: Mathematical Inductions and the Binomial Theorem. Mathematical Induction is a powerful proof technique in mathematics used to establish the truth of statements involving natural numbers. It involves two steps: the base case, where we prove that the statement is true for the first natural number, and the induction step, where we assume it to be true for an arbitrary natural number and then prove it for the next one.
This method is crucial in proving properties about integers and other mathematical structures. On the other hand, the Binomial Theorem is a fundamental result used for expanding expressions of the form (a + b)^n, where ‘n’ is a positive integer. It provides a systematic way to find the coefficients of each term in the expansion and plays a significant role in algebraic simplifications, probability theory, and combinatorics. Together, these topics equip students with powerful tools to understand and manipulate mathematical concepts and expressions in their academic journey.
Francesco Mourolico (1494-1575) introduced the method of mathematical induction, initially using it to demonstrate the equation stating that the sum of the first n positive odd integers equals n^2. He also presented various properties of integers and employed mathematical induction to prove some of these properties.
It is essential to note that even a single exception or instance where a mathematical formula doesn’t hold true is enough to invalidate it. Such a contradictory case or instance that disproves a mathematical formula or statement is referred to as a counterexample.
The validity of a formula or statement that depends on a variable belonging to a specific set is established only if it holds true for every element within that set. For example, consider the statement S(n) = n^2 – n + 41, claiming that it yields a prime number for every natural number n. Here are the values of the expression n^2 – n + 41 for the first few natural numbers:
n 1 2 3 4 5 6 7 8 9 10 11
S(n) 41 43 47 53 61 71 83 91 113 131 151
From the table, it seems that the statement S(n) is likely to be true. However, as we examine further, we discover that when n = 41, it serves as a counterexample, contradicting the claim made by the statement. Thus, we conclude that deriving a general formula from a few specific cases without proof is not a sound approach. This particular counterexample was identified by Euler (1707-1783).
Now, let’s consider another example where we aim to find the sum of the first n natural odd numbers. We calculate the initial sums to discern a pattern:
n (Number of Terms) Sum
1 ——————————- 1 = 1^2
2 —————————- 1 + 3 = 4 = 2^2
3 ————————– 1 + 3 + 5 = 9 = 3^2
4 ——————– 1 + 3 + 5 + 7 = 16 = 4^2
5 ————– 1 + 3 + 5 + 7 + 9 = 25 = 5^2
6 ———- 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6^2
We observe that each sum is the square of the number of terms included in the sum. Therefore, the following statement appears to be true:
For every natural number n,
1 + 3 + 5 + … + (2n – 1) = n^2 (where the nth term is 1 + (n – 1)^2)
However, it is impractical to verify this statement for every positive integer n due to the need for an infinite number of calculations, which is unfeasible.
This is where the method of mathematical induction comes into play. It is typically used to prove statements or formulas concerning the set of positive integers {1, 2, 3, …}. However, in some cases, it can also be applied to statements involving the set (0, 1, 2, 3, …).
Principle of Mathematical Induction
The principle of mathematical induction is summarized as follows:
If a proposition or statement S(n) for each positive integer n satisfies the following conditions:
S(1) is true, i.e., S(n) is true for n = 1, and
S(k + 1) is true whenever S(k) is true for any positive integer k,
then S(n) is true for all positive integers.
The procedure for applying mathematical induction involves:
Substituting n = 1 to demonstrate that the statement holds true for n = 1.
Assuming that the statement is true for any positive integer k, then proving that it also holds true for the next higher integer. This second condition can be established using one of two methods:
M1: Starting with one side of S(k + 1), derive the other side by using S(k).
M2: Proving that S(k + 1) can be established by performing algebraic operations on S(k).