Sequence and Series Class 12 Basic Mathematics Notes has been updated according to the latest syllabus of 2080. It means the solutions of Sequence and Series chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 4
Sequence and Series
We have already the some exercise before. So, here we will discuss about the principle of mathematical induction. Class 12 Sequence and Series chapter has 2 exercises in total. You can click on the buttons given below to view exercise-wise notes.
a. Method of Induction
We know that the product of two consecutive natural numbers is an even number. Thus if n and n + 1 be two consecutive natural numbers, then their product n(n + 1) is an even number. This is the general result. For n = 5 the product = 5.6 = 30 which is even number. This is the particular result obtained from the general one. This type of process of getting the particular result from the general one is known as the method of deduction.
Sequence and Series Class 12 Basic Mathematics Notes PDF
This PDF will provide the solutions of every question from the 2nd exercise of class 12 sequence and series. If you want the solutions of other exercises then you can select the exercise from the button given above.
Mathematical Induction
Many mathematical theorems or formulae which are complicated to prove by direct method, are proved easily by indirect method known as the mathematical induction. Its meaning will be clear from the following explanation.
Firstly, we prove that theorem or formulae for n = 1 When the theorem is true for n = 1 we shall prove that it is also true for n = 1 + 1 = 2. In the same way, we prove that it is also true for 2 + 1 = 3 and so on. Then, we conclude that the theorem or result is true for all values of n ∈ N.
The most important word used in this section is the “statement”. In this section, the resultor the formula to be proved is termed as “statement”. A statement involving the natural numberis denoted by P(n) where n∈N For example: The sum of two consecutive natural numbers is odd. This is a statement. So, it is denoted by P(n) the sum of two consecutive natural numbers is odd.
Use of Principle of Mathematical Induction to find sum
In solving a problem with the use of principle of mathematical induction, the following steps are to be used:
- Denote the given statement (i.e. the result to be proved) by P(n).
- Show that the given statement is true for n = 1 i.e. P(1) is true.
- Assume that the given statement is true for n = k i.e. assume P(k) is true.
- Show that the statement is true for n = k + 1 when it is true for n = k i.e. show that P(k + 1) is true when P(k) is true.
- Draw a conclusion that the statement is true for all n∈N.
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