BuilderBoi

Think about a week. How many days are in a week? How many hours are in each day? How many minutes are in each hour? How many seconds are in each minute?

1 x 1 triangles: 3

1 x 2 triangles: 2

1 x 3 triangles: 1

2 x 1 triangles: 3

2 x 2 triangles: 2

2 x 3 triangles: 1

3 x 1 triangles: 3

3 x 2 triangles: 2

3 x 3 triangles: 1 

So the sum is \(\color{brown}\boxed{18}\)

The formula for the sum is \({s \over {1 - r}} \), where s is the starting term, and r is the common ratio. 

The starting term is \({b \over 7} \), and the common term is \({b \over 7}\). This means that we have the equation \({{b \over 7} \over {1 - {b \over 7}}} = 1\). 

Solving for b, we find \(b = {7 \over 2} = \color{brown}\boxed{3.5}\)

\(\color{brown}\boxed{10,002}\)

https://web2.0calc.com/questions/graphing-parabolas-part-2

\(-t^2 + 8t + 10 = 12\)

\(-t^2 + 8t - 2 = 0\)

\(t = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(t = {-8 \pm \sqrt{-8^2-(4 \times -1 \times -2)} \over 2 \times -1}\)

\(t= {-8 \pm \sqrt{56} \over -2}\)

\(t = {-8 \pm 2\sqrt{14} \over -2}\)

\(t = {4 \pm \sqrt{14}}\)

\(t = \color{brown}\boxed{4 - \sqrt{14}}\)

Why don't you try it out? 

Pick any number with a remainder of 3, compute \(n^2 + 1\), and find the remainder when it is divided by 7. 

The radius of the circle is \(\text{Area} \over \text{semiperimeter}\). 

The semiperimeter is half of the perimeter, and find the area using Heron's formula: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\), where s is the semiperimeter. 

Here's a similar problem: https://web2.0calc.com/questions/help-geometry_3429#r6

Just make a list...

It would look like this: AAA, AAB, AAC, ABA, ABB, ABC, ACA, ACB, ACC, ... , CCB, CCC.