# 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 toothpicks to build 2 adjoined trapezoids and 13...

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 toothpicks to build 2 adjoined trapezoids and 13 toothpicks for 3 trapezoids.

a. If 1000 toothpicks are available, how many trapezoids will be in the last complete row?

b. How many complete rows will there be?

c. How many toothpicks will you use to construct these rows?

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### 1 Answer

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.