In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads whenever n is any non-negative integer, the numbers are the binomial coefficients, and n! denotes the factorial of n. This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to Chinese mathematician Yang Hui in the 13th century. The Persian mathematicia ...

Including:

• Binomial theorem - Newton's generalized binomial theorem
• Binomial theorem - Binomial type
• Binomial theorem - Proof inductive
• Binomial theorem - Trivia

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Isaac Newton generalized the formula to other exponents by considering an infinite series: where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not a ...

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Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - A proof, Binomial theorem - Trivia

Read more here: » Binomial theorem: Encyclopedia II - Binomial theorem - Newton's generalized binomial theorem

Isaac Newton generalized the formula to other exponents by considering an infinite series: where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not a ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

Read more here: » Binomial theorem: Encyclopedia II - Binomial theorem - Newton's generalized binomial theorem

We use mathematical induction. When n = 0, we have For the inductive step, assume the theorem holds when the exponent is m. Then for n = m + 1, (a + b)m + 1 = a(a + b)m + b(a + b)m by the inductive hypothesis

The binomial theorem can be stated by saying that the polynomial sequence is of binomial type. ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - A proof, Binomial theorem - Trivia

Read more here: » Binomial theorem: Encyclopedia II - Binomial theorem - Binomial type

When n = 1, . For the inductive step, assume it holds for m. Then for n = m + 1, (a + b)m + 1 = a(a + b)m + b(a + b)m = by the inductive hypot ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

Read more here: » Binomial theorem: Encyclopedia II - Binomial theorem - Proof inductive

The binomial theorem can be stated by saying that the polynomial sequence is of binomial type. ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

Read more here: » Binomial theorem: Encyclopedia II - Binomial theorem - Binomial type