**Determining the AP From the Given Terms and Conditions :**

Here we are going to see, how we determine an arithmetic progression from the given conditions.

**Question 1 :**

Determine the AP whose third term is 16 and 7th term exceeds the 5th term by 12.

**Solution :**

a_{3} = 16

a + 2 d = 16 ---(1)

a_{7} = a_{5} + 12

a + 6 d = a + 4 d + 12

a - a + 6d - 4d = 12

2d = 12

d = 12/2 ==> d = 6

By applying the value of d in (1), we get

a + 2(6) = 16

a + 12 = 16

a = 16 - 12

a = 4

Now we may find the arithmetic progression using the values of a and d.

General form of AP

a, a + d, a + 2d ,...........

4, 4 + 6, 4 + 2(6),.........

4, 10, 16, .................

**Question 2 :**

Find the 20^{th} term from the last term of the AP:

3, 8, 13,.........253.

**Solution :**

Common difference of the given sequence is 5

Now, we need to find the 20^{th} term from the last term of the sequence.

So, we have to consider 253 as first term of the sequence

253, 248, 243,...........,3

a = 253 d = 248 - 253 = -5 n = 20

a_{n} = a + (n - 1) d

a_{20 } = 253 + (20 - 1) (-5)

= 253 - 19 (5)

= 253 - 95

a_{20} = 158

**Question 3 :**

The sum of the 4^{th} and 8^{th} terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

**Solution :**

a a + 3d + a + 7d = 24 2 a + 10d = 24 2(a + 5d) = 24 a + 5d = 24/2 a + 5d = 12 -------(1) |
a a + 5 d + a + 9 d = 44 2 a + 14 d = 44 2(a + 7d) = 44 a + 7 d = 44/2 a + 7 d = 22 -------(2) |

(1) - (2)

a + 5 d = 12

a + 7 d = 22

(-) (-) (-)

------------

- 2d = -10

d = 5

Applying d = 5 in the (1), we get

a + 5(5) = 12

a + 25 = 12

a = 12 - 25

a = -13

By applying the values of a and d in general form, we get

a, a + d, a + 2d ,.........

-13, -13 + 5, -13 + 2(5),...........

-13, -8, -13 + 10,.......

- 13, -8, -3,.........

After having gone through the stuff given above, we hope that the students would have understood, how to determining the ap from the given terms and conditions.

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