Alan

I get the following:

Edit:  That last line should say 3ⁿ not 3n.

Could also approach this as follows:

See https://web2.0calc.com/questions/let-triangle-abc-be-equilateral-where-the-side-length#r2

"Let triangle ABC be equilateral, where the side length is 3. A point D is chosen at random inside the triangle. What is the probability that the length AD is at most 1?"

"Team A are playing against Team B in a baseball tournament. The first team to win three games wins the tournament. Team A have a probability of 2/3 of winning each game. Find the probability that they win the tournament."

See my answer at: https://web2.0calc.com/questions/help_88352#r3 

"Anty the ant is on the real number line, and Anty's goal is to get to 0. If Anty is at 1, then on the next step, Anty moves to either 0 or 2 with equal probability. If Anty is at 2, then on the next step, Anty always moves to 1. Let e1 be expected number of steps Anty takes to get to 0, given that Anty starts at the point 1. Similarly, let e2 be expected number of steps Anty takes to get to 0, given that Anty starts at the point 2. "

For x² + 1:   Let x = 2n + 1, where n is any positive integer.   Then  x² + 1 = (2n + 1)² + 1 = 4n² + 2n + 1 + 1 = 2(2n² + n + 1)  which is even, and hence composite (since x starts at 3 when n = 1).  There are therefore an infinite number of positive integers (all the odd numbers > 1) that satisfy the condition.

I agree with guest #4:

Push the two walls shown in CalculatorUser's picture outward until they lie flat on the ground. Then the "x-direction" distance between the two points is 7 + 12 + 1 or 20. The "y-direction" distance is 10 - 2 or 8. So the straight line between the points is given by Pythagoras as sqrt(20² + 8²) or 4sqrt(29)