State True/False.If the sixth term of an AP is zero then its 33rd term is three times its 15th term.

Question

Accepted Answer

Given that the 6th term of an AP is 0.Let the first term be a and the common difference be d. Formula for the

n
    
     th

term:

t
    n
   
   =a+
    
     n−1
   ×d
 Applying the formula of nth term of A.P. for 6th term.

∴
        t
        6
       
       =a+(6−1)d

⇒
        t
        6
       
       =a+5d

⇒a+5d=0

⇒a=−5d−−−−−(1)

Applying the formula of

n
    
     th

term of AP and calculate the 15th term.

∴
        t
        
         15
       
       =a+(15−1)d

⇒
        t
        
         15
       
       =a+14d−−−−−(2)

Substituting the value of a from equation (1) into equation (2):

∴
        t
        
         15
       
       =−5d+14d

⇒
        t
        
         15
       
       =9d−−−−−(3)

Applying the formula of

n
    
     th

term of AP and calculate the 33rd term.

∴
        t
        
         33
       
       =a+(33−1)d

⇒
        t
        
         33
       
       =a+32d−−−−−(4)

Substituting the value of a from equation (1) into equation (4):

∴
        t
        
         33
       
       =−5d+32d

⇒
        t
        
         33
       
       =27d

⇒
        t
        
         33
       
       =3×9d−−−−−(5)

Put

9d=
    t
    
     15

from equation (3) into equation (5):

∴
    t
    
     33
   
   =3×
    t
    
     15

Therefore, 33rd term is three times of 15th term. Hence, option (1) is correct.

State True/False.

If the sixth term of an AP is zero then its 33rd term is three times its 15th term.