Mathematics Investigation

1. PASCAL TRIANGLE
(Reymark J. Lubon)
One of the most interesting Number Patterns is Pascal’s Triangle (named after Blaise Pascal, a
famous French Mathematician and Philosopher).
Starting Point of the Investigation:
1. Build a triangle, start with “1” at the top, and then continue placing numbers below it in a
triangle pattern. Each number is the number directly above it added together. What have you
observe?
As I’ve observe the first diagonal are compose of 1 only, the second diagonal is a
counting numbers, the third diagonal is triangular numbers and the fourth diagonal has the
tetrahedral numbers. Another observation was that they are Symmetric.
2. Continue placing numbers below it and notice about the horizontal sums? Is there a pattern?
Make a conjecture out of it.
I observe that they double each time. So therefore, my first conjecture was
Conjecture No. 1: “Each horizontal sum is a power of 2 denoted by 𝟐 𝒏
.”
Proof:
20
= 1, 21
= 2, 22
= 4, 23
= 8, 24
= 16, 25
= 32 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛.
1
1
2
1
1 1
33 11

2. 3. Notice that each line is also the powers of 11.
 110
= 1 ( 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑙𝑖𝑛𝑒 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑎 "1")
 111
= 11 ( 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑙𝑖𝑛𝑒 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑎 "1" and "1")
 112
= 121 ( 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑙𝑖𝑛𝑒 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑎 "1" , "2" and "1")
But what happens with 115
, 116
, 𝑒𝑡𝑐.? Make a conjecture about it.
What happens with 115
? Simple! The digits just overlap, and then add the two digits like this:
115
=
So, the final answer is 161051.
116
=
So, the final answer is 1771561
Conjecture No. 2: “The horizontal numbers is a power of 11 denoted by 𝟏𝟏 𝒏
.”
4. For the second diagonal, what have you notice? What is the relation between the sums of the
two numbers below it? Construct a conjecture.
For the second diagonal, the square of a number is equal to the sum of the numbers next
to it and below both of those.
Examples:
32
= 3 + 6 = 9 42
= 6 + 10 = 16 52
= 10 + 15 = 25
Conjecture No. 3: “The square of a number is equal to the sum of the numbers next to it
and below both of those.”
1 51 01 0 5 1
1 6 1 0 5 1
61 52 01 5 61 1
1 7 7 1 5 6 1