Question

11. In an A.P sum of the first 2 term is 5 and sum of the last 2 term is 129. Then
find the sequence.​

Answer 1

Given :

For an A.P

The sum of first two terms = 5

The sum of last two terms = 129

To Find :

The A.P sequence

Solution :

Let The number in A.P be = a , (a+d) , (a+2d) , (a+3d) , .......

where first term = a

          common difference = d

According to question :

The sum of first two terms = 5

i.e a + (a+d) = 5

Or,   2 a + d = 5                                             .... (i)

And

The sum of last two terms = 129

i.e a + (n-2) d+ a + (n-1)d = 129

Or, 2 a + d (2 n - 3 ) = 129                            ... (ii)

Subtracting (i) from (ii), we get

[ 2 a + d (2 n - 3 ) ] - [ 2 a + d ] = 129 - 5

Or,   (2 a - 2 a ) + d ( 2 n - 3 - 1 ) = 124

Or,,  0 + d ( 2 n  - 4 ) = 124

Or,   d ( n - 2 ) = $\dfrac{124}{2}$

Or,  d ( n - 2 ) = 62

Or,  n = $\dfrac{62}{d}$ + 2

Since n has to be a non-zero positive integer.

Therefore, d can take the values 1, 2, 31 or 62

Basis on this,

For d = 1 ,

2 a + d = 5

Or, 2 a + 1 = 5

Or,  2 a = 5 - 1

Or,  2 a = 4

i.e    a = $\dfrac{4}{2}$ = 2

∴, a = 2 and d = 1

So, Sequence ,  a , (a+d) , (a+2d) , (a+3d) = 2 , (2 + 1) , (2 + 2 ×1 ) ( 2 + 3 ×1)

                                                                  = 2 , 3 , 4, 5 ,.....

Hence, The sequence are 2 , 3 , 4 , 5 , ............... Answer

Answer 2

Given :  in an ap sum of first two terms is 5 and sum of last two terms is 129

To find  :  Sequence

Solution:

Let say first two terms

a  , a + d

and last two terms

L , L - d

a + a + d  = 5

L + L - d  =  129

Adding both

2a + 2L = 134

=> a + L  = 67

2L  - 2a  - 2d   = 124

=> L  -   a  - d   = 62

=> L = 62 + a + d

a + (n-1)d  = 62 + a + d

=> (n - 2)d  = 62

62 = 1 * 62   or   2* 31    

4 possible cases

d = 1      using a + a + d = 5 => a  = 2

n = 64

Sequence is   2 , 3 , 4 ........................., 64 , 65

d = 2     using a + a + d = 5 => a  = 1.5

1.5 , 3.5  , ................................ 63.5 , 65.5

d = 31    => a  =  - 13

-13 , 18  , 49 , 80    

d = 62  => a = - 28.5

-28.5  ,  33.5 ,  95.5  

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