Logarithmically concave sequence

Okay, so imagine you have a bunch of numbers, and you put them in order from smallest to biggest. This is called a sequence.

Now, if you want to know if the sequence is logarithmically concave, that means that if you take the logarithm (which is just a fancy way of saying you figure out how many times you have to multiply a number by itself to get another number), and you add up two numbers in the sequence, and then take the logarithm of the sum, it will be smaller than the sum of the logarithms of each individual number.

Basically, this means that if you add up two numbers in the sequence, and then take the logarithm of the sum, you get a smaller number than if you just took the logarithm of each number separately and added them together.

Why is this important? Well, it turns out that logarithmically concave sequences have some really cool properties. For example, if you have a probability distribution (which is just a fancy way of saying you have a bunch of numbers that add up to 1, and each number represents the likelihood of something happening), and it's logarithmically concave, then you can use it to figure out things like the mean (which is just a fancy way of saying the average) and the variance (which is a measure of how spread out the numbers are) really easily.

So, in summary, a logarithmically concave sequence is a bunch of numbers that have a special property where if you add two of them together and take the logarithm of the sum, you get a smaller number than if you just took the logarithm of each number separately and added them together. It's important because it has some useful properties when you're working with probability distributions.