Question

The question is in the picture. The answer is 14 but I don't know how to get that.

Answer 1

Let's convert the problem into Arithmetic progression:
It would be: 5, 9, 13, .... 

Here, a = 5, d = 9 - 5 = 4
We know, S(n) = n/2 [ 2a + (n-1)d ]

Substitute the known values, 
434 = n/2 [ 2(5) + (n - 1)4 ]
434 * 2 = n [ 10 + 4n - 4 ]
868 = 10n + 4n² - 4n
= 4n² + 6n - 868 = 0

d = b² - 4ac
d = 6² - 4(4)(-868)
d = 36+13888
d = 13924

Now, roots = -b +- √d / 2a
= (-6 + √13924) / 2(4)   OR  (-6 - √13924) / 2(4)
= (-6 + 118) / 8   OR  (-6 - 118) / 8
= 112/8  OR  -124/8
= 14 OR -15.5

As number of sticks can't be in negative/decimal or fraction form, -15.5 would be fully rejected.

In short, Your Answer would be 14 [ Remaining root ]

Hope this helps!

Answer 2

The answer is 14. Because you need 5 toothpicks to build a trapezoid. And 9 toothpicks to connect 2 trapezoids together. 13 toothpicks for the 3 ones.

So, 434 / 31 = 14

14 rows of toothpicks