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**Sequence & Series**

A set of numbers whose domain is a real number is called a **SEQUENCE** and sum of the sequence is called a**SERIES**. If is a sequence, then the expression is

a series.

Those sequences whose terms follow certain patterns are called** progressions.**

**For example**

Also if f (n) = n^{2} is a sequence, then

f (10) = 10^{2} = 100 and so on.

**The nth term of a sequence is usually denoted by T _{n}**

Thus T

**There are three different progressions**

**Arithmetic Progression (A.P)****Geometric Progression (G.P)****Harmonic Progression (H.P)**

**Arithmetic Progression**

It is a series in which any two consecutive terms have common difference and next term can be derived by

adding that common difference in the previous term.

Therefore T_{n+1} - T_{n} = constant and called common difference (d) for all n âˆˆ N.

**Examples:**

1. 1, 4, 7, 10, â€¦â€¦. is an A. P. whose first term is 1 and the common difference is

d = (4 - 1) = (7 - 4) = (10 - 7) = 3.

2. 11, 7, 3, - 1 â€¦â€¦ is an A. P. whose first term is 11 and the common difference

d = 7 - 11 = 3 - 7, = - 1 - 3 = - 4.

If in an A. P.

**a** = first term,**d** = common difference = T_{n} - T**n-1****T _{n}** = nth term

Then a, a + d, a + 2d, a + 3d,... are in A.P.

nth term of an A.P.

The nth term of an A.P is given by the formula

Note: If the last term of the A.P. consisting of n terms be l , then

l = a + (n - 1) d

*Sum of n terms of an A.P*

The sum of first n terms of an AP is usually denoted by Sn and is given by the following formula:

Where â€˜l â€™ is the last term of the series.

**Ex.1 Find the series whose n ^{th} term is . Is it an A. P. series? If yes, find 101^{st} term.**

**Sol. **Putting 1, 2, 3, 4â€¦. We get T1, T2, T3, T4â€¦â€¦â€¦â€¦..

As the common differences are equal

âˆ´The series is an A.P.

**Ex.2 Find 8 ^{th}, 12^{th} and 16^{th} terms of the series; - 6, - 2, 2, 6, 10, 14, 18â€¦**

**Sol.**

**Properties of an AP**

I. If each term of an AP is increased, decreased, multiplied or divided by the

same non-zero number, then the resulting sequence is also an AP.

For example: For A.P. 3, 5, 7, 9, 11â€¦

II. In an AP, the sum of terms equidistant from the beginning and end is always same and equal to the sum

of first and last terms as shown in example below.

**III.** Three numbers in AP are taken as a - d, a, a + d.

For 4 numbers in AP are taken as a - 3d, a - d, a + d, a + 3d.

For 5 numbers in AP are taken as a - 2d, a - d, a, a + d, a + 2d.

**IV. **Three numbers a, b, c are in A.P. if and only if

2b = a + c.

and b is called Arithmetic mean of a & c

**Ex.3 The sum of three numbers in A.P. is - 3, and their product is 8. Find the numbers.**

**Sol.** Let the numbers be (a - d), a, (a + d). Then,

Sum = - 3 â‡’ (a - d) + a + (a + d) = - 3 â‡’ 3a = - 3 â‡’ a = - 1

Product = 8

â‡’ (a - d) (a) (a + d) = 8

â‡’ a (a^{2} - d^{2}) = 8

â‡’ (-1) (1 - d^{2}) = 8

â‡’ d^{2} = 9

â‡’ d = Â± 3

If d = 3, the numbers are - 4, - 1, 2. If d = - 3, the numbers are 2, - 1, - 4.

Thus, the numbers are - 4, - 1, 2 or 2, - 1, - 4.

**Ex.4 A student purchases a pen for Rs. 100. At the end of 8 years, it is valued at Rs. 20. Assuming that the yearly depreciation is constant. Find the annual depreciation.**

**Sol.** Original cost of pen = Rs. 100

Let D be the annual depreciation.

âˆ´ Price after one year = 100 - D = T_{1} = a (say)

âˆ´ Price after eight years = T_{8} = a + 7 (- D) = a - 7D

= 100 - D - 7D = 100 - 8D

By the given condition 100 - 8D = 20

8D = 80

âˆ´D = 10.

Hence annual depreciation = Rs. 10.

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