Verify that 1225 and 41 616 are simultaneously square and tr

Verify that 1225 and 41, 616 are simultaneously square and triangular numbers T_n = n^th triangular # S_m = m^th square # rightdoublearrow T_n = S_m if a # is both triangular and square rightdoublearrow 1/2 n(n+1) = m^2

Solution

1225

T_n = n(n+1)/2

n(n+1)/2 = 1225

=>n² + n - 2450 = 0

=> n² + 50n - 49n + 2450 = 0

=> n(n+50) - 49(n+50) = 0

=> (n+50) (n-49) = 0

=> n = -50 , 49

Since, n is a non-negative integer

Therefore n = 50

Hence 1225 is a triangular number.

S_m = m²

m² = 1225

=> m = 35

Hence 1225 is a square number as well.

41,616

n(n+1)/2 = 41616

=> n² + n - 83232 = 0

=> n² + 289n - 288n - 83232 = 0

=> n(n+289) - 288(n+289) = 0

=> (n+289) (n-288) =0

=> n = -289, 288

Since n is a non-negattive integer

Therefore n = 288

Hence, 41,616 is a triangular number

m² = 41,616

=> m = 204

Hence 41,616 is a square number as well