Updated: Feb 11
The problem given below is used to give students an introduction to pattern recognition. For students at the lower age group, questions such as "find the number of balls in the 6 term or the 8th term can be given. However, you may also extend this very problem to students in your Algebra 2 or Precalculus class. Let's take a look at this problem in more detail.
In the 1st term, one ball is listed. In the 2nd term three balls are listed. We find 6 balls listed for the 3rd term. Ask your students, how many balls are there in the 4th term.? If the students answer is 10. That is the correct answer. Continue in this manner for the next several terms and ask the students what pattern do they recognize that would conclude they got the correct answer. Here is a simple way to explain to the students how to answer questions a) and b).
a) Add the numbers 1 and 2. The result is 3. There are 3 balls above the 2nd term. Next, add 1 + 2 + 3. The sum of 6 is the number of balls above the 3rd term. Continue in this manner until you have counted 8 terms.
1 + 2 + 3 + 4 + 6 + 7 + 8 = 36
It's easy to arrive at answer with lower numbered terms. But, what if you are asked to find the number of marbles in the 200th term. It would be time consuming to add all the term numbers from 1 through 200. Finding a rule is a lot easier to finding large numbers of n. Here is how you could explain it to your students. Start with 10 terms:
List the numbers 1 through 10 in increasing order. Next, list the terms in decreasing order.
1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1
There are 10 columns and 2 rows. Let's call the columns groups. Add the sum in each of the 10 columns.
11 11 11 11 11 11 11 11 11 11 There are 10 groups of 11 in the array of 2 rows x 10 columns. Find the sum of all 11's.
The sum of 110 is the sum of the all the numbers in the 2 rows by 10 column array of numbers. Since we want only the sum of the 1st row, we need to divide by 2. This means 110 divided by 2 equals to 55. Therefore,
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55.
We can rewrite the statement as,
1+2+3+4+5+6+7+8+9+10 = (10)(11)/2 = 55.
To answer b)
Let n be any number, we can conclude that,
1 + 2 + 3 + 4 + ... + (n - 1) + n = n(n + 1)/2.
This can easily be proven by the principles of mathematical induction. However, that topic is covered in Number Theory. We will save that proof for another course. I hope this explains it a little where your students can learn to identify patterns.