the 17th term of an A. P is 13 times the second term. prove that the twelve term is five term the third term.

T_12 = 5 (T_3)

Step-by-step explanation

T_17 = 13 (T_2)

T_n = a + (n-1) d; where a=first term and d=common difference

T_17 = a + 16d

T_2 = a + d

Therefore,

a + 16d = 13 (a + d)

a + 16d = 13a + 13d

16d - 13d = 13a - a

3d = 12a

d = 4a

To prove that T_12 = 5 (T_3)

a + 11d = 5 (a + 2d)

a + 11d = 5a + 10d

since d = 4a

a + 11 (4a) = 5a + 10 (4a)

a + 44a = 5a + 40a

45a = 45a

Since LHS = RHS

Therefore; T_12 = 5 (T_3)