I think there are 502 that satisfy the condition.
Only the odd numbers provide a2 greater than a1. There are 1004 of them.
All these a2's are even, so a3 = a2/2 = (3a1 + 1)/2 which is greater than a1.
The corresponding a4's must alternate even and odd. The even ones will be a3/2 = (3a1 + 1)/4 which must be less than or equal to a1 (equality when a1 = 1). The odd ones must be 3(3a1 + 1)/2 + 1 = 9a1/2 + 5/2 > a1. There are 1004/2 = 502 of these.
Lie the two given lengths on top of each other. If you join them with a third length of 9 units you will have a straight line, but if you increase the length of the third length by 1 unit while keeping the ends joined you will get a triangle. Hence shortest third length is 10 units. I'm sure you can add up the lengths to get the perimeter.
"For alcohol the formula is 2,4 J/g·°C and for olive oil it is 1,65 J/g·°C. John heats up both substances using a calorimeter, first an unknown amount of alcohol and then, when the calorimeter has cooled completely, the same amount of olive oil. He heats up both liquids from 20 °C to 50 °C. How long does it take to heat up the alcohol when the heating up of the olive oil took 2,5 minutes?"
"Given is the function f(x) = -3x + 2
a. The graph of g starts from the graph of f by multiplying the x axis with the factor - 2/3 and shift it up 1/3. What is the function notation of g?
b. The graph of the function h(x) = -12x + 10 starts from the graph of f by multiplying the x axis and then shift it. With which factor do you have to multiply and how does the graph translate.
c. The graph of h can also start from the graph of f by first translating and then multiplying. Show how. "
Is this what you are looking for:
"What is the probability of rolling six standard, six-sided dice and getting six distinct numbers?"
Doesn't matter what the first one is: probability = 1
There are 5 possibilities for the second one: probability = 5/6
There are 4 possibilities for the third one: probability = 4/6
There are 3 possibilities for the fourth one: probability = 3/6
There are 2 possibilities for the fifth one: probability = 2/6
There is 1 possibility for the sixth one: probability = 1/6
Overall probability is obtained by multiplying all the above together (then reduce to lowest terms).