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The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
The solutions in terms of the original variable are obtained by substituting x 3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the original equation are
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,} where R {\displaystyle R} is a rational function of x {\displaystyle x} and a x 2 + b x + c {\textstyle {\sqrt {ax^{2}+bx+c}}} .
For example, the principal square root of 9 is 3, which is denoted by √ 9 = 3, because 3 2 = 3 • 3 = 9 and 3 is nonnegative. However raising x to the power of 0.5 using the y x key works if the number is entered as a real number with a complex part equal to zero. [ 9 ]
For example, using x,y,z as variables, and taking f to be an uninterpreted function, the singleton equation set { f(1,y) = f(x,2) } is a syntactic first-order unification problem that has the substitution { x ↦ 1, y ↦ 2 } as its only solution. Conventions differ on what values variables may assume and which expressions are considered ...
Donald Trump's proposed tariffs on Canada could increase U.S. gas prices by up to 70 cents a gallon, energy experts say.
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
The fixed rate for a 15-year mortgage is 6.02%, up 3 basis points from last week's average 5.99%. These figures are lower than a year ago, when rates averaged 7.29% for a 30-year term and 6.67% ...