Introduction to NCERT Solutions for Class 10 Maths Chapter 5
To have a better understanding of the exercise questions, NCERT class 10 maths arithmetic progression will also discuss patterns in which they are succeeding to obtain by adding a fixed number for the preceding terms,They will see about finding the nth and sum of n consecutive terms so, this knowledge can be useful in solving some daily life problems. The chapter talks about dealing with the introduction to arithmetic progression with the help of some daily life problems. There are simple and easy problems given at regular intervals to ease the concepts.
An arithmetic progression is defined as a sequence in which the difference between any two consecutive terms is constant. The difference between the consecutive terms is known as the common difference and is denoted by d.
Let’s check whether the given sequence is A.P: 1, 3, 5, 7, 9, 11. To check if the given sequence is in A.P or not, we must first prove that the difference between the consecutive terms is constant. So, d = a2– a1 should be equal to a3– a2 and so on… Here,
d = 3 – 1 = 2 equal to 5 – 3 = 2
Here we can see that the difference is common to both the terms. So, we can say that the given sequences are in arithmetic progression.
General Form of an Arithmetic Progression
Let say the terms a1 a2 a3……an are in AP. If the first term is ‘a’ and its common difference is ‘d’. Then, the terms can also be expressed as follows
1st term a1 = a
2nd term a2 = a + d
3rd term a3 = a + 2d
Therefore, we can also represent arithmetic progressions as
a, a + d, a + 2d, ……
This form of representation is called the general form of an AP.
nth term of an Arithmetic Progression
Let say the terms a1 a2 a3……an are in AP. If the first term is ‘a’ and its common difference is ‘d’. Then, the terms can also be expressed as follows,
2nd term a2 = a1+ d = a + d = a + (2-1) d
3rd term a3 = a2 + d = (a + d) + d = a + 2d = a + (3-1)d
likewise, nth term an = a + (n-1) d
Therefore, we can find the nth term of an AP by using the formula,
an = (a + (n – 1) d)
an is called the general term of an AP
Sum of n terms in an Arithmetic Progression
The sum of first n terms in arithmetic progression can be calculated using the formula given below.
then s= [(n/2) * (2a + (n – 1) d)]
Here s is the sum, n is the number of terms in AP, a is the first term and d is the common difference.
When we know the first term, a and the last term, l, of AP. Then, the sum of n terms is
Then s = [(n/2) * (a+l)]
Properties of Arithmetic Progressions
- If the same number is added or subtracted from each term of an Arithmetic Progression, then the resulting terms in the sequence are also in A.P with the same common difference.
- If each term in an Arithmetic Progression is divided or multiply with the same non-zero number then the resulting sequence is also in an A.P
- Three number x, y and z are in an Arithmetic Progression if 2y = x + z
- A sequence is an Arithmetic Progression if its nth term is a linear expression.
- If we select terms in the regular interval from an Arithmetic Progression then these selected terms will also be in Arithmetic Progression.