State True/False.If 10 times the 10th term of an AP is equal to 15 times the 15th term, then the 25th term of an AP. is zero.

Question

Accepted Answer

Given, 10 times the 10th term of an AP is equal to 15 times the 15th term.Applying the given condition:

10×
    t
    
     10
   
   =15×
    t
    
     15

-----(1)Let the first term be a and common difference be d.Formula used for nth term of A.P.:

t
    n
   
   =a+
    
     n−1
   d
   Applying the formula of

n
    
     th
   
    term of A.P for 
    
     10
    
     th
   
    term:

∴
        t
        
         10
       
       =a+(10−1)d

⇒
        t
        
         10
       
       =a+9d−−−−−(2)

Applying the formula of

n
    
     th 
   
    term of A.P for 
    
     15
    
     th 
   
   term:

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

⇒
        t
        
         15
       
       =a+14d−−−−−(3)

Applying the formula of

n
    
     th
   
    term of A.P for 
    
     25
    
     th
   
   term:

∴
    t
    
     25
   
   =a+(25−1)d−−−−(4)
 Putting the value of

t
    
     10
   
    and 
    t
    
     15

into equation (1):

∴10(a+9d)=15(a+14d)

⇒10a+90d=15a+210d

⇒−5a=120d

⇒a=−24d

⇒a+24d=0

⇒a+(25−1)d=0−−−−(5)

Putting

a+
    
     25−1
   d=
    t

25

from equation (4) in equation (5):

∴
    t
    
     25
   
   =0
 Therefore, 25th term of the AP is 0.Thus, the given statement is true.Hence, option (1) is correct.

State True/False.

If 10 times the 10th term of an AP is equal to 15 times the 15th term, then the 25th term of an AP. is zero.