6th Grade Math Textbook, Fundamentals

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1.Find the sum of the numbers from 1 through 25,000.

2.Find the sum of the odd numbers from 5 through 95.

3.Find the sum of the multiples of 3 from 3 through 300.

4.Discuss and Write Choose problem 2 or 3 and tell how you

found your answer.

page 25 for exercise sets.

Enrichment:

Sequence Sums

Objective To explore Gauss’s method for finding the sum of the first 100 counting numbers

• To apply Gauss’s method in finding sums of number sequences

Carl Friedrich Gauss (April 30, 1777–February 23, 1855) was
one of the world’s greatest mathematicians. He made many
discoveries in both pure and applied mathematics. Even as a
child, he had a strong understanding of number and function
and could do complex mental calculations.

A story about 7-year-old Carl Friedrich Gauss tells that he

quickly found the answer to a problem his teacher thought

would keep everyone in class busy for quite a while. That

problem was to find the sum of all the numbers from 1

through 100.

The task is to find this sum: 1  2  3  4  5 ... 96  97  98  99  100

You can list and add the numbers, you can use a calculator or computer, or you

can look for a pattern that might get to the solution even faster than a calculator.

Look for a pattern that might lead to the solution.

• The sum of the first and the last numbers in the sequence (1 100) is 101.


• The sum of the second and the next-to-last numbers in the sequence
  (99 2) is 101, and so on.


• There are 50 pairs of numbers with that sum in the sequence.


• To find the sum of all the numbers from 1 through 100, multiply
  the number of pairs (50) by the sum of the numbers in each of pair
  (101) to obtain 5050.

You can use a similar method to find the sums of other number sequences

if you can find pairs of numbers that have the same sum. For example,

find the sum of the even numbers from 2 through 400.

2  4  6  8  10 ... 392  394  396  398  400

• The sum of the first number and the last number in the sequence is 402.


• There are 100 pairs of numbers with that sum in the sequence.


• Multiply 402 by 100 to find the sum of the numbers in the sequence.
  So the sum of the even numbers from 2 through 400 is 40,200.