State if the following is true or false.

If a,b,c be the 1 st, 3rd
and nth terms respectively of an A.P., then the sum to n terms is

c+a
2

+

c
2

−
a
2

b−a

.

Question

State if the following is true or false.

If a,b,c be the 1 st, 3rd 
  and nth  terms respectively of an A.P., then the sum to n terms is

c+a
    2
   
   +

c
      2
     
     −
      a
      2

b−a

.

Accepted Answer

Given that,

a,b,c are the 1 st, 3rd 
  and nth terms respectively of an A.P.We know,The sum of the arithmetic progression is given by,

S
    n
   
   =
    n
    2

a+l
   .
  where,n are the number of terms.a is the first term.l is the last term.The third term of A.P can be represented as:

a
    3
   
   =a+2d
   Therefore,

a
      3
     
     =a+2d

⇒b=a+2d

⇒d=
      
       b−a
      2

Now, Substitute

b−a
    2

for d and c for

t
    n

in

t
    n
   
   =a+
    
     n−1
   d
   to obtain the value of n.

t
      n
     
     =a+
      
       n−1
     d

⇒c=a+
      
       n−1

b−a
        2

⇒c−a=
      
       n−1

b−a
        2

⇒2
      
       c−a
     =
      
       n−1

b−a

⇒n−1=
      
       2
        
         c−a

b−a

⇒n=1+
      
       2
        
         c−a

b−a

⇒n=
      
       b−a+2c−2a

b−a

⇒n=
      
       b−3a+2c
      
       b−a

Again, Substitute

b−3a+2c
    
     b−a

for n and c for l in

S
    n
   
   =
    n
    2

a+l
   
   to obtain the sum.

S
      n
     
     =
      n
      2

a+l