Arithmetic Progression is one the most important and scoring Topics in class tenth mathematics. In this article, we will learn the basic concepts of arithmetic progression.
An Arithmetic progression can be defined as a sequence of numbers in which the consecutive terms (beginning with the second term) are formed by adding a constant quantity with the preceding term. Examples of arithmetic expression are –
- Natural number – 1,2,3,4,5…. With 1 as first term as well as the constant quantity.
- 2,4,6,8,10…. It is an AP with 2 as first term as well as the constant quantity.
- 4,4,4,4,4,4…. It is an AP with 4 as first term and 0 as the constant quantity.
- A taxi charges Rs 16 for first km and Rs 10 each additional km. Fare would be 16, 26, 36….
- Price of a commodity is Rs 80. Each year its price decreases by Rs 10.
Resulting AP would be 80, 70, 60, 50… where 80 is the first term and -10 is the constant quantity.
Finite Arithmetic Progression can be defined as the AP with finite number of terms. For example, AP – 2,4,6,8,10 has 5 terms.
Infinite Arithmetic Progression can be defined as the AP with infinite number of terms. For example, AP – 3,6,9,12…. has infinite number of terms.
First term of an Arithmetic Progression is denoted by symbol ‘a’. Common Difference(constant quantity) is denoted by symbol ‘d’.
General form of AP is – a, a+d, a+2d, a+3d….
To determine the value of a term at a certain position in the AP, we have the formula for nth term of an AP.
nth term of AP, an = a + (n-1) d
Sum of first n terms, Sn, can be determined by
Sn = n/2 [2a + (n-1) d]
Also, if the nth of the last term, l, is already known then sum of first n terms would be
Sn = n/2 [a + l]
Arithmetic mean is the quantity obtained by summing two or more numbers or variables and then dividing by the number of numbers or variables. For example, if a,b,c are in AP, then b = [a+c]/2 and b would be called as arithmetic mean of a and c.