eudigm arithmetic progression

ARITHMETIC PROGRESSION

Consider the sequence 2 , 5 , 8 , 11,.

What do you observe ?


The first term of the sequence is 2.

Anything else ?


Each and every term of the sequence is separated from the other term by a common difference of 3.

Let us go through some examples of this type of sequence


Ex. 1 : 20 , 16 , 12 , 8,..

Ex .2 : 3 , 23 , 33 , 43,..

Ex.3 : - 3, 82 , 167, 252 , .


A sequence is called an A.P. if the difference f a term and its predecessor is always constant. This constant term is called the common difference of the arithmetic progression


Take an A.P. a , a + d , a + 2d , , a + ( n -1 ) d

What is the first term ? What is the common difference ?


Two types of A.P.

1. Finite A.P.

2. Infinite A.P.


Check the following sequences

1. 1.2 , 3.2 , 4.5 ,

2. 2 , 8 , 18 ,

3. -45,44,133 , 222,..

4. -48,37,74,..


More Examples 1. Write first three terms of the A.P. where first

term and common difference are 7 and 9 respectively.

2. Write first four terms of the A.P. where first term and common difference are 3 and 3 respectively

3. Write first five terms of the A.P. where first terms and common difference are -22 and 45 repsectively


More Examples Continued

4. Write first four terms of the A.P. where first term and common difference are 14.5 and 20.5 respectively

5. Write first four terms of the A.P. where first term and common difference are -5 and 20.3 respectively


General Term of an A.P.

If the first term of an A.P. is a ,

second term a+d ,

third term a+2d ,

and so on then the nth term is a+(n-1) d


EXAMPLES

1. In the A.P. 2,4,6, find the 8th term

2. In the A.P. -2,2,6,10, find the 40th term

3. In the A.P. 2,22,32, find the 18th term

4. In the A.P. -0.5 , 0.5,1.5, find the 60th term


Sum of First n Terms of an A.P.

Let us consider an A.P. a , a+d , a+2d , .

Then the sum of the first n terms of the A.P. would be (n/2) [ 2a + ( n -1 ) d ]


EXAMPLES

1. Find the sum of the first 20 terms of the A.P. 5,9,13,17,22,.

2. Find the sum of the first 50 terms of the A.P. whose 2nd term is 2 and 4th term is 8

3. How many terms of the A.P. : 24 , 21 , 18 , must be taken so that their sum is 78 ?


Arithmetic Mean

If a,b,c are in A.P. then b-a=c-b

Therefore b = (a+c)/2 is called the arithmetic mean of a and c

eudigm arithmetic progression

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