Session 10: Progressions, BODMAS, Functions | A NEW OUTLOOK – CSIR-NET Series

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Arithmetic Sequence

In an arithmetic sequence, each term of a sequence differs from its preceding term by a constant called the common difference.

For Example: 1, 4, 7, 10, 13, 16, 19, …

If a is the first term and d is the common difference,

• The n^th term is, t_n = a + (n – 1)d
• Sum of n terms of an A.P. =

 (a)  To select odd number of terms

• a – d, a, a + d    (three terms)
• a – 2d, a – d, a, a + d, a + 2d  (five terms)
• a – 3d, a – 2d, a – d, a1a + d, a + 2d, a + 3d  (seven terms)

(b)  To select even number of terms

•  a – d, a + d  (two terms)
• a – 3d, a – d, a + d, a + 3d  (four terms)

Geometric Sequence

In a geometric sequence, every term bears a constant ratio with its preceding term.

For Example: 2, 6, 18, 54, …

• The nth term of a geometric series, t_n = ar^{n – 1}
• Sum of n terms of a Geometric sequence = a(rⁿ – 1)/(r -1) (where a is the first term, r is the common ratio and r > 1)
• Sum of terms of an infinite geometric progression = a/(1-r)   (when r < 1)

Function

A function defines a relation between two sets – associating an object from one set to another unique object in another set.

For Example: Let f(x) = x², where x is from the set of all real numbers. Then f(3) = 32 = 9.

Constant Function

A function given by f(x) = c, where c is a fixed number is called a constant function.

For Example: Let D = { a, b } and E = { 1, 2, 3 }, then the function f:D → E given by f(x) = 2 is a constant function.

Identity Function

A function given by f(x) = x, is called the identity function.

BODMAS rule

Bracket – of – Division – Multiplication – Addition – Subtraction

Depicts the correct sequence in which the operations are to be executed, to find out the value of a given mathematical expression.

Thus, in simplifying an expression, first of all the brackets must be removed (by solving the expressions in the brackets) strictly in the order ( ), { }, and [ ].

After removing the brackets, we must use the following operations strictly in the order, and priority given to the one that occurs first from the left:

(i) Division or Multiplication

(ii) Addition or Subtraction

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