![]() ![]() Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand. This gives us any number we want in the series. I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:į(x) = 17.5x^2 - 27.5x + 15. Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic. ![]() = a ( 4 ) + 2 =a(4)+2 = a ( 4 ) + 2 equals, a, left parenthesis, 4, right parenthesis, plus, 2 = 9 =\goldD9 = 9 equals, start color #e07d10, 9, end color #e07d10Ī ( 5 ) a(5) a ( 5 ) a, left parenthesis, 5, right parenthesis = 7 + 2 =\blueD 7+2 = 7 + 2 equals, start color #11accd, 7, end color #11accd, plus, 2 = a ( 3 ) + 2 =a(3)+2 = a ( 3 ) + 2 equals, a, left parenthesis, 3, right parenthesis, plus, 2 = 7 =\blueD 7 = 7 equals, start color #11accd, 7, end color #11accdĪ ( 4 ) a(4) a ( 4 ) a, left parenthesis, 4, right parenthesis = 5 + 2 =\purpleC5+2 = 5 + 2 equals, start color #aa87ff, 5, end color #aa87ff, plus, 2 = a ( 2 ) + 2 =a(2)+2 = a ( 2 ) + 2 equals, a, left parenthesis, 2, right parenthesis, plus, 2 = 5 =\purpleC5 = 5 equals, start color #aa87ff, 5, end color #aa87ffĪ ( 3 ) a(3) a ( 3 ) a, left parenthesis, 3, right parenthesis = a ( 1 ) + 2 =a(1)+2 = a ( 1 ) + 2 equals, a, left parenthesis, 1, right parenthesis, plus, 2 = 3 =\greenE 3 = 3 equals, start color #0d923f, 3, end color #0d923fĪ ( 2 ) a(2) a ( 2 ) a, left parenthesis, 2, right parenthesis = a ( n − 1 ) + 2 =a(n\!-\!\!1)+2 = a ( n − 1 ) + 2 equals, a, left parenthesis, n, minus, 1, right parenthesis, plus, 2Ī ( 1 ) a(1) a ( 1 ) a, left parenthesis, 1, right parenthesis As a result, all input language facilities as well as the underlying manipulation routines may be interactively extended by an experienced user.A ( n ) a(n) a ( n ) a, left parenthesis, n, right parenthesis Otherwise, evaluation of expressions occurs in the current environment created by the successive user commands, with certain operations such as integration, differentiation, and simplification performed automatically.Translators for the user language and for a resident higher-level procedural language facility are written in META/LISP, a new self-compiling translator-writing system. The user may also enter syntax definition statements in order to introduce new notations into the system.Expressions appearing in assignment statements may include "where"-clauses which allow user control over the "environment" used in evaluation. Assignment statements are the fundamental commands in the user language they may contain "for"-clauses which restrict the domain for which the assignment is valid and permit "piecewise" and recursive definition of new operators and functions. Finding Missing Numbers To find a missing number, first find a Rule behind the Sequence. Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion. Data objects include sequences (both finite and infinite) and arrays of arbitrary rank. A Sequence is a set of things (usually numbers) that are in order. Using this LISP system as a base, portions of several systems have been combined and augmented to provide the following facilities to a user:(1) rational function manipulation and simplification symbolic differentiation (Anthony Hearn's REDUCE)(2) symbolic integration (Joel Moses' SIN)(3) polynomial factorization, solution of linear differential equations, direct and inverse symbolic Laplace transforms (Carl Engelman's MATHLAB, including Knut Korsvold's simplification system)(4) unlimited precision integer arithmetic(5) manipulation of arrays containing symbolic entries(6) two-dimensional output on IBM 2741 terminals or IBM 2250 displays (William Martin's Symbolic Mathematical Laboratory, and Jonathan Millen's CHARYBDIS program from MATHLAB)(7) self-extending language facility (META/LISP).The user language created for the system incorporates a subset of "customary" mathematical notation. ![]() To carry out this objective, an experimental LISP system has been implemented for IBM System/360 computers. This paper describes a system designed to provide an interactive symbolic computational facility for the mathematician user. ![]()
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