DETERMINING THE AP FROM THE GIVEN TERMS AND CONDITIONS

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 20th 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 20th 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

an  =  a + (n - 1) d

a20  =  253 + (20 - 1) (-5)

  =  253 - 19 (5)

  =  253 - 95

a20  =  158  

Question 3  :

The sum of the 4th and 8th 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 :

a4 + a8  =  24  ----(1)

a + 3d + a + 7d = 24

2 a + 10d = 24

2(a + 5d) = 24

a + 5d = 24/2

a + 5d = 12 -------(1)

 a6 +  a10  =  44

 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,......... 

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