Mathematical induction: Principle, AM & GM
Principle of Mathematical Induction
THE NATURAL NUMBERS are the counting numbers: 1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement — a theorem, or a formula — that is asserted about every natural number.
By “every”, or “all,” natural numbers, we mean any one that we name.
1 + 2 + 3 + . . . + n = ½n(n + 1).
This asserts that the sum of consecutive numbers from 1 to n is given by the formula on the right. We want to prove that this will be true for n = 1, n = 2, n = 3, and so on. Now we can test the formula for any given number, say n = 3:
1 + 2 + 3 = ½· 3· 4 = 6
— which is true. It is also true for n = 4:
1 + 2 + 3 + 4 = ½· 4· 5 = 10.
But how are we to prove this rule for every value of n?
The method of proof is the following. We show that if the statement — the rule — is true for any specific number k (e.g. 104), then it will also be true for its successor, k + 1 (e.g. 105). We then show that the statement will be true for 1. It then follows that the statement will be true for 2. Therefore it will be true for 3. It will be true for any natural number we name.
This is called the principle of mathematical induction.
1) When a statement is true for a natural number n = k,
then it will also be true for its successor, n = k + 1;
2) The statement is true for n = 1;
then the statement will be true for every natural number n.
To prove a statement by induction, we must prove parts 1) and 2) above.
The hypothesis of Step 1) — “The statement is true for n = k” — is called the induction assumption, or the induction hypothesis.
In general language arithmetic mean is same as the average of data. It is the representative value of the group of data. Suppose we are given ‘ n ‘ number of data and we need to compute the arithmetic mean, all that we need to do is just sum up all the numbers and divide it by the total numbers.
Question: The runs scored by Sachin in 5 test matches are 140, 153, 148, 150 and 154 respectively. Find the mean.
Solution: The correct option is B. Runs scored by Sachin in 5 test matches: 140, 153, 148, and 154
Means of the runs = total runs / number of matches
Mean = (140+153+148+150+154)/5 = 745/5 = 149
Question: Mean of a set of observations is the value which
- Occurs most frequently
- Divides observation into two equal parts
- Is a representative of the whole number.
- Is the sum of observations
Solution: The correct option is C. Mean is the value which is the representative of the whole number. It takes into account of all the values present in the group and averages them.
The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as “the nth root product of n numbers.” The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.
The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is it takes into account the effects of compounding.
The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.
Example: What is the Geometric Mean of 2 and 18?
- First we multiply them: 2 × 18 = 36
- Then (as there are two numbers) take the square root: √36 = 6
In one line:
Geometric Mean of 2 and 18 = √(2 × 18) = 6
Example: What is the Geometric Mean of 10, 51.2 and 8?
- First we multiply them: 10 × 51.2 × 8 = 4096
- Then (as there are three numbers) take the cube root: 3√4096 = 16
In one line:
Geometric Mean = 3√(10 × 51.2 × 8) = 16