If a series has finite number of terms then it is called a Finite series. If a series has infinite number of terms then it is called an Infinite series. Let us focus our attention only on studying finite series.

The sum of the terms of
a sequence is called series. Let*a*_{1}, *a*_{2}, *a*_{3},...,
*a _{n}* ,... be the sequence of real numbers. Then the real number

If a series has finite
number of terms then it is called a Finite series. If a series has infinite number of terms then
it is called an Infinite series. Let us focus our attention only on studying finite series

A series whose terms are
in Arithmetic progression is called Arithmetic series.

Let *a* , *a*
+ *d* , *a* + 2*d* , *a* + 3*d*,... be the Arithmetic
Progression.

The sum of first *n*
terms of a Arithmetic Progression denoted by *S _{n}* is given by,

*S _{n}=a* +(

Rewriting the above in
reverse order

*S _{n}* =(

Adding (1) and (2) we
get,

2*S* * _{n}* =[

= [2*a* +(*n*
− 1)*d*
] +[2*a* + (*n*
−1)*d*
+ .. .. +[2*a* + (*n* −1)*d*
] (*n*
terms)

2*S _{n}*=

**Note**

If the first term** ***a*,
and the last term** ***l*** **(*nth*** **term) are
given then

**Example 2.31 **Find the sum of first** **15** **terms of the A. P.** **

*Solution*

Here the first term* **a*** **=

Sum of first *n*
terms of an A.P.

Find the sum of** **0. 40** **+** **0. 43** **+** **0. 46** **+ + 1 .

Here the value of* **n*** **is not given. But the
last term is given. From this, we can

Given *a* = 0. 40
and *l* = 1 , we find *d*= 0. 43 −
0.40 = 0. 03 .

Therefore,

Sum of first *n*
terms of an A.P. *S _{n} = *

Here, *n* = 21
. Therefore,

So, the sum of 21 terms
of the given series is 14.7.

How many terms of the
series**
**1** **+** **5** **+** **9** **+** **...** **must be taken so that
their** **sum is 190?

Here we have to find the
value of*
**n*, such that* **S _{n}*

First term *a* = 1,
common difference *d* = 5 −1 = 4 .

Sum of first *n*
terms of an A.P.

But *n* = 10 as *n*
= −19/2 is impossible. Therefore, *n* = 10 .

State True or False.
Justify it.

1. The *n*^{th}
term of any A.P. is of the form *pn*+*q* where *p* and *q*
are some constants.

2. The sum to *n*^{th}
term of any A.P. is of the form *pn*^{2}+*qn* + *r*
where *p*, *q*, *r* are some constants.

The** **13*th*** **term of an A.P. is** **3** **and the sum of first** **13** **terms is** **234. Find the** **common difference and
the sum of first 21 terms.

Given the** **13

Sum of first 13 terms =
234 gives *S* _{13 }=

** **2

Solving (1) and (2) we get, *a* = 33, *d*
= −5 /2

Therefore, common
difference is -5/2

Sum of first 21 terms = *S _{21} *

In an A.P. the sum of
first** ***n*** **terms is** **5*n*^{2}/2** **+** **3*n/*2** **. Find the** **17^{th}** **term.** **

** Solution **The

Find the sum of all
natural numbers between** **300** **and** **600** **which are** **divisible by 7.

The natural numbers
between**
**300

The sum of all natural
numbers between 300 and 600 is 301 + 308 + 315 + + 595.

The terms of the above
series are in A.P.

First term *a* =
301 ; common difference *d* = 7 ; Last term *l* = 595.

Since, *S _{n}*
=

A mosaic is designed in
the shape of an** **equilateral triangle, 12 ft on each side. Each tile in the mosaic
is in the shape of an equilateral triangle of 12 inch side. The tiles are
alternate in colour as shown in the figure. Find the number of tiles of each
colour and total number of tiles in the mosaic.

Since the mosaic is in the shape of an equilateral triangle of 12 ft, and the tile is in the shape of an equilateral triangle of 12 inch (1 ft), there will be 12 rows in the mosaic

From the figure, it is
clear that number of white tiles in each row are 1, 2, 3, 4, …, 12 which
clearly forms an Arithmetic Progression.

Similarly the number of
blue tiles in each row are 0, 1, 2, 3, …, 11 which is also an Arithmetic
Progression.

Number of white tiles =
1 + 2 + 3 + … + 12 = 12/2
[1 + 12] = 78

Number of blue tiles = 0
+ 1 + 2 + 3 + … + 11 = 12/2 [0 + 11] = 66

The total number of
tiles in the mosaic = 78 + 66 = 144

The houses of a street
are numbered from** **1** **to** **49. Senthil’s house is** **numbered such that the sum of numbers of the
houses prior to Senthil’s house is equal to the sum of numbers of the houses
following Senthil’s house. Find Senthil’s house number?

Let Senthil’s house
number be* **x*.* *

It is given that

*x ^{2}* =2450 −

*x ^{2}* =1225 gives

Therefore, Senthil’s
house number is 35.

**Example 2.39**

The sum of first** ***n*, 2*n*** **and** **3*n*** **terms of an A.P. are** ***S*_{1}** **,*S*_{2}** **and** ***S*_{3}** **respectively.

Prove that *S*_{3}
= 3(*S*_{2} −*S*_{1} ).

If* **S*_{1}** **,

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