TECHNIQUES FOR ADDING THE NUMBERS 1 TO 100

     

“What is the sum of the first one hundred numbers beginning from one?” (i.e.

*
?)

After the teacher asked this question, the class, which was full of young kids, fell into complete silence. A few students were stunned by the seemingly-impossible challenge and readily gave up; most students began scribbling on the paper, trying to địa chỉ all the numbers one by one, from the very beginning. What a difficult question! They thought.

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But there was one kid in the class did it differently. He thought about it for a few minutes, did some simple calculation, & raised his hand.

“I am finished,” the student said.

“How is it possible,” the teacher said lớn himself as he walked toward the student, “the problem would take one at least an hour khổng lồ do!” Indeed, if he had solve the problem himself, he would just sum up all the one hundred numbers one by one as well – as a matter of fact, he presented the problem khổng lồ the class just to lớn kill some time. But after he examined his student’s answer, he was shocked.

“It’s a genius’ solution!” after a few seconds of freezing in astonishment, the teacher shouted, “this kid is going to be famous!”

It was in the late 18th century, Germany. The teacher was right – it turned out lớn be that his brilliant student, Johann Carl Friedrich Gauss, became one of the most famous and important mathematicians of all time.

So, how did young Gauss bởi the calculation?

First, he wrote the sum twice, one in an ordinary order and the other in a reverse order:

1 + 2 + 3 + 4 + . . . + 99 + 100100 + 99 + . . . + 4 + 3 + 2 + 1

By adding vertically, each pair of numbers adds up khổng lồ 101:


1

+

2

+

3

+

. . .

+

98

+

99

+

100

100

+

99

+

98

+

. . .

+

3

+

2

+

1


Since there are 100 of these sums of 101, the total is 100 X 101 = 10,100. Because this sum 10,100 is twice the sum of the numbers 1 through 100, we have:

*

Shake Hands with Arithmetic Sequences!

It is believed that Gauss was the first person who discovered this beautiful method of finding such a sum. Furthermore, he also gave a generalization to the problem and thus opened the gate to lớn a whole new world in algebra – the world of Arithmetic Sequence.

Put in very simple terms, an Arithmetic Sequence is a certain number of numbers arranged in a way such that the difference between any two consecutive numbers is the same. You see, there are two words underlined. They are the từ khoá of our definition: the total number of terms (or numbers. We use terms and numbers interchangeably hereafter) we have, and the common difference between terms.

So for example, we can say that young Gauss was facing a sequence, really, which can be written as:

*

And we know, the first term is 1, the last term is 100, the number of terms is 100, và the common difference (the difference between any two consecutive terms. Sometimes called the constant difference) is 1.

Pair ‘Em Up và Sum!

As you see, the key idea is Gauss’s method is lớn pair the numbers up. It turns out to lớn be that this “Pairing Up” idea is one of the most important ideas in mathematics, & they often yield lớn very beautiful solutions.

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*

Pair up, numbers, & let the dance begin! Pair-wise summation techniques, when used with insight, often yield lớn beautiful solutions.


There are several ways we can pair numbers up:

Rewrite all numbers backward underneath the original sequence and pair up the numbers vertically (This was Gauss’ approach).Pair the first term & the last term of the sequence, the second và the second-last, the third and third-last … proceed until you reach the middle of the sequence và can go no further. In this method, we do not have khổng lồ write the sequence again, & this is the good news. The bad news, however, is that you have to keep an eye on whether if there is a middle term, the one term in the very center that must be left alone (for example, in the sequence 1, 2, 3, 4, 5, the middle term would be 3). Sometimes, it is a good idea to modify all numbers in a sequence beforehand in order to lớn make our calculation even easier. We will see this technique in some examples later on.Can you think of more ways? địa chỉ cửa hàng them here!

Now let’s attempt some exercises. Don’t worry if you get stuck on some of those questions – they are meant to lớn be challenging. Just start scribbling on your scratch paper & think hard!

Exercise 1

Calculate:

*

Exercise 2

Calculate:

*

Exercise 3

Calculate:

*

Exercise 4

In a pyramid-shaped building, there are 22 stories. There are 30 people on the tenth floor, và each other floor has exactly one more person than the floor above it. So in total how many people are there in the building?

Exercise 5

In a party, there are ten friends. At the end of the party, the friends shake hands with each other lớn bid farewell. Each individual shakes hand to every other one exactly once. How many handshakes took place? (hint: if we number the friends as No.1, No.2, …, No.10, how many handshakes did No.1 make? How many NEW handshakes did No. 2 make? How about the others?)

Exercise 6

The sum of the first twenty-five numbers is 325. What is the sum of the next twenty-five numbers?

Exercise 7

Consider the sequence: 1, 4, 7, 10, … .What is the 10th term in the sequence? What is the sum of the first ten terms?

Exercise 8

One day, Tom is very bored và he begins khổng lồ record the hours his digital clock displays. He writes down the hour whenever an exact hour occurs (for example, he would write down number 17 when it is 5PM sharp). He starts at 8AM in the morning, và finishes at 3AM the next morning, when he goes to lớn bed. What is the sum of all the numbers he writes down? note that Tom’s clock shows 00:00 when midnight.

Exercise 9

Consider the following tower of numbers:

1

2 3

4 5 6

7 8 9 10

. . .

What is the second number from the left in the 10th row from the top? What is the third number from the right in the 17th row?

Exercise 10

Now let’s try khổng lồ write the formula for the sum of an arithmetic sequence! Can you use only the terms “first term”, “last term”, “common difference” và “number of terms” khổng lồ express the sum of the first few terms of the sequence?


“What is the sum of the first one hundred numbers beginning from one?” (i.e. 1 + 2 + 3 + 4 + . . . + 100 =?)

After the teacher asked this question, the class, which was full of young kids, fell into complete silence. A few students were stunned by the seemingly-impossible challenge & readily gave up; most students began scribbling on the paper, trying to địa chỉ cửa hàng all the numbers one by one, from the very beginning. What a difficult question! They thought.

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But there was one kid in the class did it differently. He thought about it for a few minutes, did some simple calculation, và raised his hand.

“I am finished,” the student said.

“How is it possible,” the teacher said khổng lồ himself as he walked toward the student, “the problem would take one at least an hour khổng lồ do!” Indeed, if he had solve the problem himself, he would just sum up all the one hundred numbers one by one as well – as a matter of fact, he presented the problem to lớn the class just khổng lồ kill some time. But after he examined his student’s answer, he was shocked.

“It’s a genius’ solution!” after a few seconds of freezing in astonishment, the teacher shouted, “this kid is going khổng lồ be famous!”

It was in the late 18th century, Germany. The teacher was right – it turned out lớn be that his brilliant student, Johann Carl Friedrich Gauss, became one of the most famous và important mathematicians of all time.

So, how did young Gauss bởi vì the calculation?

First, he wrote the sum twice, one in an ordinary order và the other in a reverse order: