The sum of all the solutions in 0,100 for the equation sin ⁡ π x + cos ⁡ π x = 0 is

Step-by-step solution

sin ⁡ π x + cos ⁡ π x = 0 tan ⁡ π x = - 1 = t a n 3 π 4 π x = n π + 3 π 4 ⇒ x = n + 3 4 Given, x ∈ 0,100 , hence n will range from 0 to 99 . x will be 3 4 , 7 4 , 11 4 , . . . . . , 399 4 S n = n 2 2 a + n - 1 d = 50 6 4 + 99 × 1 = 502 5