<h3>
Answer: 625</h3>
Work Shown:
The set {A,B,C,1,2} has five items. There are four slots to fill.
So we have 5^4 = 5*5*5*5 = 625 different possible passwords where the characters can be repeated.
Let's start by visualising this concept.
Number of grains on square:
1 2 4 8 16 ...
We can see that it starts to form a geometric sequence, with the common ratio being 2.
For the first question, we simply want the fifteenth term, so we just use the nth term geometric form:
![T_n = ar^{n - 1}](https://tex.z-dn.net/?f=T_n%20%3D%20ar%5E%7Bn%20-%201%7D)
![T_{15} = 2^{14} = 16384](https://tex.z-dn.net/?f=T_%7B15%7D%20%3D%202%5E%7B14%7D%20%3D%2016384)
Thus, there are 16, 384 grains on the fifteenth square.
The second question begs the same process, only this time, it's a summation. Using our sum to n terms of geometric sequence, we get:
![S_n = \frac{a(r^{n} - 1)}{r - 1}](https://tex.z-dn.net/?f=S_n%20%3D%20%5Cfrac%7Ba%28r%5E%7Bn%7D%20-%201%29%7D%7Br%20-%201%7D)
![S_{15} = \frac{2^{15} - 1}{2 - 1}](https://tex.z-dn.net/?f=S_%7B15%7D%20%3D%20%5Cfrac%7B2%5E%7B15%7D%20-%201%7D%7B2%20-%201%7D)
![S_{15} = 2^{15} - 1 = 32767](https://tex.z-dn.net/?f=S_%7B15%7D%20%3D%202%5E%7B15%7D%20-%201%20%3D%2032767)
Thus, there are 32, 767 total grains on the first 15 squares, and you should be able to work the rest from here.
Answer:
D. 2
Step-by-step explanation:
(1 - -5)/(4 - 1) = 6/3 = 2
1. First, do 12 x 8 to work out the area of the rectangle, which is 96ft.
Then, to work out the area of a circle, you use the equation πr² to help you. You would multiply π by the radius², which is 16. Now you have just worked out the area of a circle, but not a semicircle, so you would have to divide your answer by two to get the area of this, which would be 25.13 (rounded to 2 d.p).
To get the area of the whole shape, you just have to add the two totals together.
25.13 + 96 = 121.13ft.
Remember to put the units there, or you can lose marks.
Try your best with the next questions! I have written the formulas for the other shapes to help you work out the answers.
Area of a square = Multiply sides together.
Area of rectangle = Multiply width by length.
Area of a circle = Multiply π by the radius².
Area of a semicircle = Multiply π by the radius², and the divide by two.
Area of a triangle = Multiply the base by the height, and then divide by two.
Really hope this helps!