State if the following is true or false.

If the sum of first m terms of an A.P. is the same as the sum of its first n terms, then the sum of its first (m + n) terms is zero.

Question

State if the following is true or false.If the sum of first m terms of an A.P. is the same as the sum of its first n terms, then the sum of its first (m + n) terms is zero.

Accepted Answer

We are given that the sum of first m terms of an A.P. is the same as the sum of its first n terms.We know,The sum of the arithmetic progression is,

S
    n
   
   =
    n
    2

2a+
      
       n−1
     d
   .
  The first term is a.The common difference is d.The number of terms are n.Let the first term be a and common difference of the given A.P is d. The sum of the first m terms can be represented as

S
    m
   
   =
    m
    2

2a+
      
       m−1
     d
   
   and the sum of the n terms can be represented a

S
    n
   
   =
    n
    2

2a+
      
       n−1
     d
   .
  Now, Equate the equation

S
    m
   
   =
    m
    2

2a+
      
       m−1
     d
   
   and

S
    n
   
   =
    n
    2

2a+
      
       n−1
     d
   
   to obtain the equation in terms of a, m, n and d.

S
      m
     
     =
      S
      n

⇒
      m
      2

2a+
        
         m−1
       d
     =
      n
      2

2a+
        
         n−1
       d

⇒2am+md
      
       m−1
     =2an+nd
      
       n−1

⇒2am−2an+
      m
      2
     
     d−md−
      n
      2
     
     d+nd=0

⇒2a
      
       m−n
     +d

m
          2
         
         −
          n
          2

−
        
         m−n
       
     =0

⇒2a
      
       m−n
     +d

m−n

m+n
       −
        
         m−n
       
     =0

⇒
      
       m−n

2a+

m+n
         −1
       d
     =0

⇒2a+
      
       m+n−1
     d=0

But

m−n≠0
   .We need to substitute 0 for

2a+
    
     m+n−1
   d
   in

S
    
     m+n
   
   =
    
     m+n
    2

2a+

m+n
       −1
     d
   
   to obtain the sum of the m and n terms of A.P.

S
      
       m+n
     
     =
      
       m+n
      2

2a+

m+n
         −1
       d

⇒
      S
      
       m+n
     
     =
      
       m+n
      2

0

⇒
      S
      
       m+n
     
     =0

Hence proved the sum of its first

m+n
   
   terms is zero,

S
    
     m+n
   
   =0.
  Therefore, the statement is true.Hence, option 1 is correct.

State if the following is true or false.

If the sum of first m terms of an A.P. is the same as the sum of its first n terms, then the sum of its first (m + n) terms is zero.

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