Coupon Accepted Successfully!


Arithmetic Progression (A. P)

Consider the sequences;
  1. 2, 6, 10, 14, 18, ......
  2. -9, -5, -1, 3, 7, ......
We observe that in the above sequences, each term differs from the preceding term by a constant quantity. These types of sequences are called Arithmetic Sequences or Arithmetic Progression.

Now consider the sequence: 2, 3, 5, 7, 11, 13,..... There is no pattern (formula) by which the next number can be found out, except that we know the next prime number is 17. This example is not a progression but a sequence.

All progressions are sequences; but all sequences are not progressions.

Let us recall what we have already learnt about A.P.
  1. is the standard form of an A.P, whose first term is and common difference is
  2. The term (general term) of an A.P is
  3. If is the last term (term of an A.P), then
  4. Sum to terms of an A.P (denoted by ) is given by

Some properties of A.P

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

Example 1:
The term of an A.P is twice the term. Prove that


Example 2:
If are in A.P., Prove that are also in A.P.

Given are in A.P.


Example 3:


Example 4:


Example 5:


Example 6:
An A.P has terms. Prove that n times the sum of the odd terms is equal to times the sum of the even terms.

Let be the A.P. Out of terms,

Hence the result.

Example 7:
The sum of terms of an A.P is (). Find the common difference.


Example 8:
The ratio of the sums of m and n terms of an A.P is . Show that the ratio of


Example 9:
If the sum of terms of an A.P is term is 164, find the value of .


Example 10:
The difference between any two consecutive interior angles of a polygon is . If the smallest angle is , find the number of sides of the polygon.

Let n be the number of sides.

The polygon can have either 9 sides or 16 sides.

Test Your Skills Now!
Take a Quiz now
Reviewer Name