Can you solve these sums of Gaussians from 1 to 10?
Puzzles that require some thought and knowledge provide an enjoyable challenge. It can help improve your skills as you solve these puzzles. These questions are about adding numbers, and they get you thinking because they require you to use your problem-solving skills. But one thing to remember – you need to think about patterns and how the numbers fit together. Additionally, some studies show that solving such puzzles can help keep your skills sharp.
The sum of Gaussians is a mathematical concept named after the distinguished German mathematician Johann Carl Friedrich Gauss. He is considered one of the most influential mathematical thinkers in history. There is a famous story about him that at a very young age he came up with an ingenious way to add numbers. His math teacher once asked the class to add all the numbers from 1 to 100. Adding them up one by one is a lot of numbers, right? The teacher may have thought it would take a long time, but young Gauss quickly came up with the answer: 5050.
Now, this story may not be entirely true, but it is a reminder that sometimes young students can discover new mathematical ideas. Gauss found a clever way to group numbers between 1 and 100 so that they are easier to add. Let’s examine a problem to understand how this grouping technique works.
You want to add numbers between 1 and 10
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ?
So think about solving mathematical puzzles like Gauss.
Can you solve these sums of Gaussians from 1 to 10?solution
If you want to add numbers between 1 and 10. You do this by matching up the numbers. First, add the first number to the last number, which is 1 + 10. Then, add the second number to the second-to-last number, which is 2 + 9. Continue to follow this pattern.
So you have:
(1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) = ?
Each pair adds up to 11. So, you can think about our problem this way:
(11) + (11) + (11) + (11) + (11) = ?
Since we have 5 pairs, our answer is:
11 + 11 + 11 + 11 + 11 = 11 x 5 = 55
Try the next method. Instead of arranging the numbers in one row, you can arrange the numbers in two rows. In the first row, the numbers increase from 1 to 10. In the second row, the number decreases from 10 to 1. When you add each column together, you get 11 for each column. To find the sum of all numbers, you multiply the number of pairs by the sum of each pair. But we only want the sum of one row, not the sum of two rows. Therefore, we divide the answer by 2.
The formula is as follows: sum = (number of pairs x sum of pairs) / 2. For our example, it is (10 x 11) / 2, which equals 55.
Try using other methods to add numbers. In algebra, we can represent this pattern using the letter “n” to represent how many numbers there are in the list. For our example, “n” is 10. The number of pairs is “n” divided by 2. You’ll notice that the size of a pair is the number of pairs plus 1. Therefore, we can write it as:
(logarithm) x (sum of each pair) = (n/2) x (n + 1), which is equal to (10 x 11) / 2, which is also equal to 55.
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