Arithmetic progression( A.P)

Sometimes we get these types of number series viz 1,3,5,7,9,........ Or 2,4,6,8,10.....or 3,6,9,12,15,....or 10,20,30,40,50,......and so on. Here, if we closely observe the number series then we find that the differences between the first term and second term is same with the one of the second term and third term. The difference is same with all the pair of the numbers. If you notice the first example then the difference between 1 and 3 is 2, 3 and 5 is also 2, 7  and 9 is also 2. In the second example the difference is also 2 between the pairs of numbers. In the third example the difference is 3 between the pairs of numbers. In the fourth example the difference is 10.

These type of arrangement of numbers is called arithmetic progression. So when the differences between the number of pairs is same then it is called arithmetic progression or A.P.

The  difference is called common difference and it is denoted by d. The first term of the series is denoted by a.

So first remember the abbreviations that are used in arithmetic progression. Now there are some formulae that are used in solving the problems relating A.P.

First nth term=tₙ=a+(n-1)×d

Where a=first term, n=no of terms, d=common difference.

Example1: what is the twentieth(20th) term of the series 3,7,11,15,.....?

Ans: (first of all we will try to find all the values that are in the formula and substitute these values to the formula)

Here first term=a=3

Common difference=d=second term-first term

                                        =7-3=4

nth term=n=20

The 20th term=a+(20-1)×d

                         =3+(20-1)×4

                         =3+19×4

                         =3+76

                         =79.

Second formula is summation formula:

 Sₙ= (n÷2)×{2a+(n-1)× d} 

where a=first term,  d=common difference, n= number of term.

Example 2:  what is the sum up to 30 terms of the series 5,7,9,11,....

Ans: here a=5, d=7-5=2, n=30

So S(30)=(30÷2)×{2×5+(30-1)×2}

                =15×{10+29×2}

                =15×{10+58}

                =15×68

                =1020.