If the sum of n terms of an A.P. is np + 12 n( n —1)q , where p and q are constants, then the common difference of the A.P is ____.

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If the sum of n terms of an A.P. is np+ 12 n( n —1)q , where p and q are constants, then the common difference of the A.P is q.Given that the sum of n terms of an A.P. is np + 12 n( n —1)q .We know,The sum of n terms of an AP is given by

S
    n
   
   =
    n
    2

2a+
      
       n−1
     d
   
  . The nth term of an AP is given by

a
    n
   
   =a+
    
     n−1
   d
  .Where,The first term is a.The common difference is d.The number of terms are n.Substitute 1 for n in

S
    n
   
   =np+
    1
    2
   
   n
    
     n−1
   q
   to obtain the value of

a
    1

S
      1
     
     =
      1
     p+
      1
      2

1

1
       −1
     q

⇒
      S
      1
     
     =p+
      1
      2
     
     ×0

⇒
      S
      1
     
     =p

So,

p=
    a
    1

Substitute 2 for n in

S
    n
   
   =np+
    1
    2
   
   n
    
     n−1
   q
   to obtain the  value of

S
    2

S
      2
     
     =
      2
     p+
      1

2

2
       −1
     q

⇒
      S
      2
     
     =2p+q

Subtract the  value of

S
    1

from

S
    2

to obtain the value of

a
    2

,

a
      2
     
     =
      S
      2
     
     −
      S
      1

⇒
      a
      2
     
     =
      
       2p+q
     −
      p

⇒
      a
      2
     
     =2p+q−p

⇒
      a
      2
     
     =p+q

Subtract value of

a
    1

from

a
    2

to obtain the value of d,d=a2-a1⇒d=(p+q)-(p)⇒d=p+q-p⇒d=qThe common difference is q.

If the sum of n terms of an A.P. is np + 12 n( n —1)q , where p and q are constants, then the common difference of the A.P is ____.

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