Question

. Let tn denote the number of integral sided triangle with distinct sides chosen from {1, 2, 3, , , ,n). Then

t20 - t10 equals​

Answer 1

Answer:

t20 - t10  = 470

t20 - t19 = 81

Step-by-step explanation:

as sides are integral

Hence with Shortest side = 1 no triangle can exist

then

side = 2  , 17 Triangles can exists  

(3 , 4) , (4 , 5) ..................................................(19 , 20)

Side = 3  

(4 , 5) ( 4, 6) , (5 , 6) , (5 , 7)............................................(18 , 19) , (18 , 20) , (19 , 20)

31 Triangles

Side = 4

(5 , 6) , (5 , 7) , (5 , 8) .............................................................(17 , 18),(17, 19) , (17 , 20) , (18 , 19) , (18 , 20) , (19 , 20)

= 42 Triangles

Side = 5    = 50 triangles

Side = 6   = 55 triangles

Side = 7  = 57 triangles

Side = 8  = 56  triangles

Side = 9  = 52 triangles

Side = 10 = 45  triangles

Side = 11  =  36  triangles

Side 12  = 28   triangles

Side 13 = 21    triangles

Side 14  = 15   triangles

Side 15  = 10   triangles

Side 16  = 6    triangles

Side 17  = 3   triangles

Side 18 = 1    triangles

Total Triangles = 17 + 31 + 42 + 50 + 55 + 57 + 56 + 52 +45 + 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1

= 525

Similarly  for n = 10

7 + 11  + 12 + 10 + 6 + 3 + 1 = 55 Triangles

t20 - t10  = 525 - 55 = 470

t19

16 + 29 + 39 + 46 + 50 + 51 + 49 + 44 + 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1

Similarly t19 = 444

t20 - t19 = 525 - 444 = 81  Triangles

Formula derived by induction :

(a - 1) (n + 1 - 2*a)  + (a - 2)(a-1)/2  ( this formula hold true  until 2a < n + 1)

then it beomes  (n - a)(n-a-1)/2

a = shortest side