Using principle of mathematical induction prove the following
To prove a statement using the principle of mathematical induction, we typically have to follow these steps:
1. Base Step: Show that the statement holds true for the base case (usually n = 0 or n = 1).
2. Inductive Step: Assume that the statement holds true for some arbitrary positive integer k. Then, using this assumption, prove that the statement also holds true for the next value (k + 1).
Now, let's use the principle of mathematical induction to prove a statement.
Statement: For all positive integers n, the sum of the first n positive integers is given by the formula: 1 + 2 + ... + n = n(n + 1)/2.
Step 1: Base Step
When n = 1, the left-hand side (LHS) of the equation is 1, and the right-hand side (RHS) is (1)(1 + 1)/2 = 1, which are equal. Hence, the statement holds true for the base case.
Step 2: Inductive Step
Assume that for some positive integer k, i.e., the sum of the first k positive integers is given by the formula: 1 + 2 + ... + k = k(k + 1)/2.
Now, we need to prove that the statement also holds true for the next value (k + 1).
Consider the sum of the first (k + 1) positive integers, i.e., 1 + 2 + ... + k + (k + 1).
Using the assumption from the inductive step, we can rewrite this as:
[k(k + 1)/2] + (k + 1).
Simplifying this expression, we get:
[(k^2 + k)/2] + (k + 1).
Combining the terms in the numerator, we have:
[(k^2 + k + 2k + 2)/2] = [(k^2 + 3k + 2)/2] = [(k + 1)(k + 2)/2].
Therefore, the sum of the first (k + 1) positive integers is given by the formula: (k + 1)(k + 2)/2.
Since we assumed the formula to be true for k and showed that it holds true for (k + 1), by the principle of mathematical induction, the statement is true for all positive integers.
Hence, the formula 1 + 2 + ... + n = n(n + 1)/2 is proven using the principle of mathematical induction.