String Art the TI-Nspire way! The instructions are on page 1 of the file.
The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to Gauss approached with his answer: The teacher suspected a cheat, but no.
Manual addition was for suckers, and Gauss found a formula to sidestep the problem: Pair Numbers Pairing numbers is a common approach to this problem. As the top row increases, the bottom row decreases, so the sum stays the same.
And how many pairs do we have? Wait — what about an odd number of items? What if we are adding up the numbers 1 to 9? Many explanations will just give the explanation above and leave it at that. However, our formula will look a bit different. If you plug these numbers in you get: Yep, you get the same formula, but for different reasons.
Use Two Rows The above method works, but you handle odd and even numbers differently. The total of all the numbers above is But we only want the sum of one row, not both. So we divide the formula above by 2 and get: Now this is cool as cool as rows of numbers can be.
It works for an odd or even number of items the same! Make a Rectangle I recently stumbled upon another explanation, a fresh approach to the old pairing explanation. Different explanations work better for different people, and I tend to like this one better.
Instead of writing out numbers, pretend we have beans. We want to add 1 bean to 2 beans to 3 beans… all the way up to 5 beans. How do we count the number of beans in our pyramid?
The next row of the pyramid has 1 less x 4 total and 1 more o 2 total to fill the gap. Just like the pairing, one side is increasing, and the other is decreasing. Now for the explanation: How many beans do we have total? If we have numbers 1…then we clearly have items.
To get the average, notice that the numbers are all equally distributed. Even though we have a fractional average, this is ok — since we have an even number of items, when we multiply the average by the count that ugly fraction will disappear.
Notice in both cases, 1 is on one side of the average and N is equally far away on the other. So, we can say the average of the entire set is actually just the average of 1 and n: Putting this into our formula And voila!
We have a fourth way of thinking about our formula. So why is this useful? Notice that the formula expands to this: Having a firm grasp of this formula will help your understanding in many areas.
By the way, there are more details about the history of this story and the technique Gauss may have used. Variations Instead of 1 to n, how about 5 to n? Just double the regular formula.Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration).
The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.. The power of recursion evidently lies in the possibility of defining an infinite set of objects by a. If sum of factorial of each digit of the number is same as the actual number then that number is called as strong number.
A simple solution is to find all prime factors of both numbers, then find intersection of all factors present in both numbers.
Finally return product of elements in the intersection. An efficient solution is to use Euclidean algorithm which is the main algorithm used for this purpose.
The idea is, GCD of two numbers doesn’t change if smaller number is . Back to top A cell is a flexible type of variable that can hold any type of variable. A cell array is simply an array of those cells. It's somewhat confusing so let's make an analogy.
A cell is like a bucket. You can throw anything you want into the bucket: a string, an integer, a double, an. Because of Dan Bricklin and Bob Frankston's implementation of VisiCalc on the Apple II in and the IBM PC in , the spreadsheet concept became widely known in the late s and early s.
VisiCalc was the first spreadsheet that combined all essential features of modern spreadsheet applications (except for forward referencing/natural . OK, let us now focus on to the main logic section.
Below is the step by stive logic to find sum of first 10 odd numbers. Initialize variable that will hold the sum.