PINKLETES
Pascal's Triangle
Pascal's Triangle is a triangular array of numbers named after the French mathematician Blaise Pascal. The triangle is constructed such that each number in the triangle is the sum of the two numbers directly above it.
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Binomial Expansion
A binomial expression is the sum, or difference, of two terms.
For example: a+b 2x+y 7-b
Pascal's Triangle is used in expanding binomial expressions raised to positive integer powers. Each row in the triangle corresponds to the coefficients of the binomial expansion of (a + b)^n
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Example:
Notice that the coefficients 1 2 1 match the values on the 2nd row, that is equal to the power (2)
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Example:
Notice that the coefficients of each term directly correspond to the third rows values
1 3 3 1
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Additionally, you may notice that variable a's exponent starts at N and then decreases to 0, while variable b's exponent starts at 0 and then increases to N.
Power of 11
Each line represents a power of 11^n, starting with n=0 at the first row.
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11^0 = 1
11^1 = 11
11^2 = 121
11^3 = 1331
11^4 = 14641
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When n>4 the numbers become double digits and it becomes more difficult to solve for the powers of 11^n.
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For example: n=5 1 5 10 10 5 1
You can solve this by multiplying each term by 10^a, as a progressively increased by 1 starting from 0
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1(10^5)+5(10^4) + 10(10^3)+ 10(10^2) +5(10^1) +1(10^0) = 100000 + 50000 + 10000 + 1000 + 50 + 1
11^5 = 161051
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Fibonacci Sequence
The fibonacci sequence is a sequence in which number is the sum of the two numbers before it
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0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
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The fibonacci sequence can be spotted as the sum of the diagonals.
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Starting from 0, if a row starts with a prime number or is a prime numbered row, all the numbers that are in that row (not counting the 1's) are divisibly by that prime numbers.
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For example:
- Row and prime number 7 is divisible by all numbers not counting 1
- Row and composite number 6 is not divisible by 15 and 20
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Prime Numbers
The sum of each row of pascal's triangle can be calculated with
2^n, where n is the row number -1
For example:
The sum of the 4th row is 1 + 3 + 3 + 1 = 8 = 2^3