If 1000 toothpicks are available, how many trapezoids will be in the last complete row?It takes 5 toothpicks to build the top trapezoid. You need 9...

Let   the top single trapezoid which has
5

So the 1st row   has 5 tooth pics.

a1
= 4*1+1.

In the 2nd row , there would be 2 adjoined trapezoids and
has a2 = 5+4= 4*2+1 = 9 tooth pics.

In the 3rd row the re are 3
reapezoids with 3*4+1 = 13.

Like that in the nth row there are an
=4n+1  toothpicks

 Therefore the sum Sn of the tooth picks of n rows
is given by:

Sn = a1+a2+a3+...+an

Sn =
(1*4+1)+(2*4+1)+(3*4+1) +....(n*4+1)

Sn =
(1+2+3+...+n)4+(1+1+1....+1), there are n 1's in the 2nd bracket.

Sn
= n(n+1)4/2 +n = 2n(n+1)+n = 2n^2+3n = 1000.

2n^2+3n
=1000.

2n^2+3n > 1000.

For n =
21,  n(2n+3) = 945.

fOr n =22, n(2n+3) =
1034.

So there are 21 rows.

The last
complete row has 21 trapezoids with 21*4 +1 = 85 tooth picks.

The
last 23rd incomplete row has 13 trapezoids with 13*4+1 = 53
sticks

Totatal tooth pickes used = 21(21*4+1) = 945+13*4+1  =
998.