# Arithmetic Progression (A. P)

Consider the sequences;

- 2, 6, 10, 14, 18, ......
- -9, -5, -1, 3, 7, ......

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.

**Note:**

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

Let us recall what we have already learnt about A.P.

- is the standard form of an A.P, whose first term is and common difference is
- The term (general term) of an A.P is
- If is the last term (term of an A.P), then
- 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

**Solution:**

**Example 2:**

If are in A.P., Prove that are also in A.P.

**Solution:**

Given are in A.P.

**Note:**

**Example 3:**

If

**Solution:**

**Example 4:**

**Solution:**

**Example 5:**

**Solution:**

**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.

**Solution:**

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.

**Solution:**

**Example 8:**

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

**Solution:**

**Example 9:**

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

**Solution:**

**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.

**Solution:**

Let n be the number of sides.

âˆ´ The polygon can have either 9 sides or 16 sides.