Revision Notes for Ch 5 Arithmetic Progression Class 10th Maths

Arithmetic Progression

Arithmetic Progression: An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.

Term: The fixed number d is called the common difference of the A.P.

Common Difference: Each number in the list of arithmetic progression is called term. It can be positive, negative or zero.

General form of an AP: If a is first term and d is common difference of an A.P. The general form of an AP is: a, a+d, a+2d, a+3d
(iii) ax3 + bx+ cx + d is  a polynomial of degree 3 cubic polynomial.

Finding General Term of an AP: We can find the nth term of an A.P by using Tn = a + (n-1)d
We can find the required term by putting the numeric value of n(no. of term) which we have to find from starting.

nth term from the end: l - (n-1)d where, l is the last term

The number occurring at nth place is denoted by Tn . The first term is T, second term is Tand so on.
Common difference d = T n-1 - T n

Examples:-
3, 7, 11, 15,19………….
This is an arithmetic progression which starts from 3. The common difference between two consecutive terms is 4.
That is
7-3=4, 11-7=4, 15-11=4
So, this is an AP with first term as 3, and common difference 4.
Or, we can say the AP is
3, 3+4, 3+4+4=3+2×4, 3+4+4+4=3+3×4
That is 4 is added to each term to get the next term.

Arithmetic Mean

If a,A, b are in AP we say that A is arithmetic mean between a and b. It is also written as AM
Let A is the AM between two numbers a and b
Let A be the AM between a and b, then 
As we know that common difference is difference  between two consecutive term of an AP, and it is same between two consecutive terms of the number. 
So,
⇒A-a=b-A
⇒2A=b+a
A=(a+b)/2
So, arithmetic mean of a and b is (a + b)/2.

Sum of n-terms of an AP

Sum of n term of an AP is

Here n is number of terms of AP, a is first term and l is last term of the AP.  Here, d is the common difference. 

Properties of Arithmetic Progression

If a constant is added or subtracted or multiplied to each term of an A.P. then the resulting sequence is also an A.P.
If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.
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