# Write a recursive formula for an arithmetic sequence

Site Navigation Arithmetic Sequences This lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference. If you need to review these topics, click here. This is primarily a list of Greatest Mathematicians of the Past, but I use birth as an arbitrary cutoff, and two of the "Top " are still alive now. Click here for a longer List of including many more 20th-century mathematicians.

Click for a discussion of certain omissions. Please send me e-mail if you believe there's a major flaw in my rankings or an error in any of the biographies. Obviously the relative ranks of, say Fibonacci and Ramanujan, will never satisfy everyone since the reasons for their "greatness" are different.

I'm sure I've overlooked great mathematicians who obviously belong on this list. Please e-mail and tell me! Following are the top mathematicians in chronological birth-year order. By the way, the ranking assigned to a mathematician will appear if you place the cursor atop the name at the top of his mini-bio.

Earliest mathematicians Little is known of the earliest mathematics, but the famous Ishango Bone from Early Stone-Age Africa has tally marks suggesting arithmetic.

## Recursion (computer science) - Wikipedia

The markings include six prime numbers 5, 7, 11, 13, 17, 19 in order, though this is probably coincidence. The advanced artifacts of Egypt's Old Kingdom and the Indus-Harrapa civilization imply strong mathematical skill, but the first written evidence of advanced arithmetic dates from Sumeria, where year old clay tablets show multiplication and division problems; the first abacus may be about this old.

By years ago, Mesopotamian tablets show tables of squares, cubes, reciprocals, and even logarithms and trig functions, using a primitive place-value system in base 60, not Babylonians were familiar with the Pythagorean Theorem, solutions to quadratic equations, even cubic equations though they didn't have a general solution for theseand eventually even developed methods to estimate terms for compound interest.

The Greeks borrowed from Babylonian mathematics, which was the most advanced of any before the Greeks; but there is no ancient Babylonian mathematician whose name is known.

Also at least years ago, the Egyptian scribe Ahmes produced a famous manuscript now called the Rhind Papyrusitself a copy of a late Middle Kingdom text. It showed simple algebra methods and included a table giving optimal expressions using Egyptian fractions.

Today, Egyptian fractions lead to challenging number theory problems with no practical applications, but they may have had practical value for the Egyptians.

The Pyramids demonstrate that Egyptians were adept at geometry, though little written evidence survives. Babylon was much more advanced than Egypt at arithmetic and algebra; this was probably due, at least in part, to their place-value system.

## What is a Sequence?

But although their base system survives e. The Vedics understood relationships between geometry and arithmetic, developed astronomy, astrology, calendars, and used mathematical forms in some religious rituals. The earliest mathematician to whom definite teachings can be ascribed was Lagadha, who apparently lived about BC and used geometry and elementary trigonometry for his astronomy.

Apastambha did work summarized below; other early Vedic mathematicians solved quadratic and simultaneous equations. Other early cultures also developed some mathematics. The ancient Mayans apparently had a place-value system with zero before the Hindus did; Aztec architecture implies practical geometry skills.

Ancient China certainly developed mathematics, in fact the first known proof of the Pythagorean Theorem is found in a Chinese book Zhoubi Suanjing which might have been written about BC.

Thales may have invented the notion of compass-and-straightedge construction. Several fundamental theorems about triangles are attributed to Thales, including the law of similar triangles which Thales used famously to calculate the height of the Great Pyramid and "Thales' Theorem" itself: The other "theorems" were probably more like well-known axioms, but Thales proved Thales' Theorem using two of his other theorems; it is said that Thales then sacrificed an ox to celebrate what might have been the first mathematical proof in Greece.

Thales was also an astronomer; he invented the day calendar, introduced the use of Ursa Minor for finding North, invented the gnomonic map projection the first of many methods known today to map part of the surface of a sphere to a plane, and is the first person believed to have correctly predicted a solar eclipse.

His theories of physics would seem quaint today, but he seems to have been the first to describe magnetism and static electricity. Aristotle said, "To Thales the primary question was not what do we know, but how do we know it. It is said he once leased all available olive presses after predicting a good olive season; he did this not for the wealth itself, but as a demonstration of the use of intelligence in business.

Thales' writings have not survived and are known only second-hand. Since his famous theorems of geometry were probably already known in ancient Babylon, his importance derives from imparting the notions of mathematical proof and the scientific method to ancient Greeks.

Thales' student and successor was Anaximander, who is often called the "First Scientist" instead of Thales: Anaximander is famous for astronomy, cartography and sundials, and also enunciated a theory of evolution, that land species somehow developed from primordial fish!

Anaximander's most famous student, in turn, was Pythagoras. The methods of Thales and Pythagoras led to the schools of Plato and Euclid, an intellectual blossoming unequaled until Europe's Renaissance.

For this reason Thales may belong on this list for his historical importance despite his relative lack of mathematical achievements.

Apastambha ca BC India The Dharmasutra composed by Apastambha contains mensuration techniques, novel geometric construction techniques, a method of elementary algebra, and what may be an early proof of the Pythagorean Theorem.The recursive formula for an arithmetic sequence is written in the form For our particular sequence, since the common difference (d) is 4, we would write So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence.

Write a recursive formula for the arithmetic sequence below. What is the value of the 8th term? 1, 5, 9, 13,/5(2). PROBLEM POSING. Students have asked me, on several occasions, "Is there any math after calculus?" These students have been given the impression that the world of mathematics is both finite and linear (the classic algebra-through-calculus sequence).

Arithmetic Sequences. Reviewing common difference, extending sequences, finding the nth term, finding a specific term in an arithmetic sequence, recursive formula, explicit formula. STUDY. PLAY. sequence. number pattern.

## Arithmetic Sequences

arithmetic sequence. Write the first five terms of the sequence, explain what the fifth term means in context to the. This is an introduction to R (“GNU S”), a language and environment for statistical computing and graphics.

R is similar to the award-winning 1 S system, which was developed at Bell Laboratories by John Chambers et al. It provides a wide variety of statistical and graphical techniques (linear and. For example, find the recursive formula of 3, 5, 7, If you're seeing this message, it means we're having trouble loading external resources on our website.

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Making Mathematics: Mathematics Research Teacher Handbook