Mathematical Induction

Mathematical induction is a process which can be applied to set of general statements for sets of positive integers or their connected sequences.

In this we find either a given statement is true for entire natural numbers or not. As we discussed that in the case of natural numbers we will include only positive numbers. This should not be interpreted as a form of inductive reasoning. The easiest form of induction characterizes that either a statement that includes a natural number 'n' holds for all values of 'n' is true or not.

It can be understood with help of an example:

Example: Show that sum of first 'n' odd integers is n²?

It can be written as 1 + 3 + 5 + 7 + ... + (2n - 1) = n² for all positive integers.

Let’s have a proof of above expression: First find the value of A (n). It is equals to A (n): 1 + 3 + 5 + 7 + ... + (2n - 1) = n², then in basic step we have to show A (1) is true. So it can be written as: Trivial: 1 = 1².

In Inductive step we need to show that A (n) is true for all 'n' therefore A (n + 1) is also true for all 'n'. Let A (n) is true so 1 + 3 + 5 + 7 + ... + (2n - 1) = n²,

Also write it as A (n + 1): 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)² follows:

 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)²,

It can also be written as:

n² + (2n + 1) = (n + 1)², this is the proof of mathematical induction.

Precipitation Reaction is used in formation of solid in a solution or inside another solid.

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