DETERMINING THE AP FROM THE GIVEN TERMS AND CONDITIONS

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.

Determining the AP From the Given Terms and Conditions - Examples

Question 1 :

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

Solution :

a₃ = 16

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

 a₇ = a₅ + 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

a₂₀  =  253 + (20 - 1) (-5)

  =  253 - 19 (5)

  =  253 - 95

a₂₀  =  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 :

(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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