Binomial Theorem what are its functions and what it means
- 1. Understanding Pascal's Triangle Exploring its Patterns and Properties
- 2. Exploring the Fascinating Pascal's Triangle Pascal's triangle reveals patterns in binomial coefficients. Pascal's triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)n. It is named for the 17th-century French mathematician Blaise Pascal, but it is far older.
- 3. Exploring Pascal's Triangle Basics A pascal's triangle is an arrangement of numbers in a triangular array such that the numbers at the end of each row are 1 and the remaining numbers are the sum of the nearest two numbers in the above row. This concept is used widely in probability, combinatorics, and algebra.
- 4. Exploring the Fascinating Pascal's Triangle Understanding its patterns and mathematical significance. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is the numbers directly above it added together. (Here I have highlighted that 1+3 = 4)
- 5. Patterns Within the Triangle Diagonals The first diagonal is, of course, just "1"s The next diagonal has the Counting Numbers (1,2,3, etc). The third diagonal has the triangular numbers (The fourth diagonal, not highlighted, has the tetrahedral numbers.)
- 6. Symmetrical The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image. Horizontal Sums What do you notice about the horizontal sums? Is there a pattern? They double each time (powers of 2). (Why? Because each number in the current row is used twice to make the next row.)
- 7. Horizontal Sums What do you notice about the horizontal sums? Is there a pattern? They double each time (powers of 2). (Why? Because each number in the current row is used twice to make the next row.)
- 8. Exponents of 11 Each line is also the powers (exponents) of 11: > 110=1 (the first line is just a "1") >111=11 (the second line is "1" and "1") > 112=121 (the third line is "1", "2", "1") etc!
- 9. But what happens with 115 ? Simple! The digits just overlap, like this:
- 10. Odds and Evens If we color the Odd and Even numbers, we end up with a pattern the same as the Sierpinski Triangle
- 11. The same thing happens with 116 etc. Squares 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, ...
- 12. Fibonacci Sequence Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence. (The Fibonacci Sequence starts "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)
- 13. BINOMIAL THEOREM
- 14. The binomial theorem formula is (a+b)n= ∑nr=0nCr an-rbr, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n. This formula helps to expand the binomial expressions such as (x + a)10, (2x + 5)3, (x - (1/x))4, and so on. What is binomial theorem
- 15. What is a binomial coefficient with an example? In combinatorics, the binomial coefficient is used to denote the number of possible ways to choose a subset of objects of a given numerosity from a larger set. It is so called because it can be used to write the coefficients of the expansion of a power of a binomial.
- 16. Binomial theorem expansion formula The binomial theorem formula is (a+b)n= ∑nr=0nCr an-rbr, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n. This formula helps to expand the binomial expressions such as (x + a)10, (2x + 5)3, (x - (1/x))4, and so on.
- 17. If a and b are real numbers and n is a positive integer, then (a + b)n =nC0 an + nC1 an – 1 b1 + nC2 an – 2 b2 + ... 1. The total number of terms in the binomial expansion of (a + b)n is n + 1, i.e. one more than the exponent n. What is the formula for the binomial theorem combination?