# Method of Mathematical Reasoning and Aptitude for Paper 1 Unit V (Mathematical Reasoning and Aptitude) as Per New 2021 Syllabus

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## Mathematical Reasoning and Aptitude: Mathematical Induction

- The word ‘induction’ means to generalization a statement from some given facts.
- The principle of mathematical induction is a techniques to establish the truth of a given mathematical statement which has been formulated on terms of the natural number ‘n’ .
- Assume that the statement is p (n) which has an association with some positive integer ‘n’ .
- Step 1 involves an examination of the statement for n = 1
- In step 2 it is assumed that the statement is true for a positive integer ‘k’ .
- In step 3, the truth of p (k + 1) is established.

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## Deductive Reasoning

- Statement
- Any statement is an acceptable mathematical statement if it is either a true statement or a false statement, but it cannot be both. It is generally denoted as an alphabet followed by a colon & then the statement.

- Compound statement
- A combination of multiple mathematical statements through some logical.

- Negation of a statement
- If any statement is denied being true, then that is the statement.
- If a statement is denoted by ‘p’ then negation is denoted by ‘~p’ .

## Logical Operations

- The compound statements are connected by some logical operations. These logical operations are denoted by some special phrases, also called as connectives.
- ‘And’ operation
- If each of the component statement in a compound statement is ‘true’ then the compound statement is ‘true’ . Even if any one of the statement using is ‘false’ , the compound statement is ‘false’ . It is denoted by ‘v’ .

- ‘Or’ operation
- If anyone (or more than one) component statement of a compound statement is/are true, then the compound statement is ‘true’ . The compound statement using ‘or’ is ‘false’ only if all the component statements are false. It is denoted by ‘^’ .

- Implies
- A compound statement ‘a’ implies ‘b’ means that the statement ‘a’ is enough condition for statement ‘b’ & vice-versa. It is denoted by p is q.

- ‘And’ operation

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# Mathematical Reasoning and Aptitude: How to Identify Common Patterns in Number Series

- The common patterns asked in number series topic are-
- Arithmetic series
- This could be of two types. In which the next term is obtained by adding/subtracting a constant number to its previous term. Common key is the key here
- Example: 4, 9, 14, 24,29, 34 [in the given series pattern is continued by adding 5 to the last number each time]
- Furthermore this could be mixed with arithmetic with the differences of consecutive numbers themselves form an arithmetic series.
- Example: 1, 3, 6,10, 15 … .
- Example: 1, 3, 6,10, 15 … . [In the given series 3 – 1 = 2,10 – 6 = 4,15 – 10 = 5 … . Now, we get an arithmetic sequence 2,3, 4,5]
- Geometric sequences is multiplying by the same value each time.

- Prime numbers
- Squares/cubes
- Pattern in differences
- Pattern in alternate numbers
- As the name of the series specifies, this type of series may consist of two series combined into a single series. The alternating terms of this series may form an independent series in itself.
- Example: 3, 4, 8, 10,13, 16? ?
- As we can see, there are two series formed ⇾ series 3,8, 13 with a common difference of 5
- ⇾ series 2: 4,10, 16 with a common difference of 6
- So, next two terms of the series should be 18 22 respectively.

- Squares/cubes

- Arithmetic series