Let T _ r be the r ^ th term of an A.P. If for some m , T _ m = 1 25 , ~T ₂₅= 1 20 , and 20 _ r =1 ²⁵ ~T _ r =13 , then 5 ~m _ r = m ^ 2 ~m ~T _ r is equal to

Step-by-step solution

aligned & T _ m = 1 25 , ~T ₂₅= 1 20 , 20 _ r =1 ²⁵ ~T _ r =13 & ~T _ m = a +( m -1) d = 1 25 .(1) & T ₂₅= a +24 ~d = 1 20 & 20 25 2 [ a + 1 20 ]=13 a = 1 500 aligned also, 20 ~S ₂₅=20 25 2 [2 a +24 ~d ]=13 ~d = 1 500 from (1) 1 500 + m -1 500 = 1 25 ~m =20 Now, 5 ~m _ r = m ^ 2 ~m ~T _ r =100 _ r =20 ⁴⁰ ~T _ r =126