Next 1.4

When faced with a sequence for which you need to find missing values or the next few values, you need first to look at it and see if you can get a 'feel' for what is going on. For instance: Find the next number in the following sequence: 1, 4, 9, 16, 25. 5 International Drive, Suite 210 Rye Brook, New York. 28470 Avenue Stanford, Suite 200 Valencia, California. In Jumanji: The Next Level, the gang is back but the game has changed. As they return to rescue one of their own, the players will have to brave parts unknown from arid deserts to snowy mountains, to escape the world's most dangerous game. Non-Mathematical 'Sequences' (page 6 of 7) Sections: Common differences, Recursions, General examples, Non-math 'sequences' There are quite a few sequences handed out to students in math classes that are not even mathematical sequences. NextNumber finds the next number in a sequence of numbers Find next number. About NextNumber. Classic Sequences. Contact NextNumber. Classic Sequences. Contact. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + is an elementary example of a geometric series that converges absolutely. There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2 −1 + 2 −2 + 2 −3 +.

First six summands drawn as portions of a square.

The geometric series on the real line.

In mathematics, the infinite series1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely.

There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2⁻¹ + 2⁻² + 2⁻³ + ...

The sum of this series can be denoted in summation notation as:

{displaystyle {frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots =sum _{n=1}^{infty }left({frac {1}{2}}right)^{n}={frac {frac {1}{2}}{1-{frac {1}{2}}}}=1.}

Proof[edit]

As with any infinite series, the infinite sum

{displaystyle {frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots }

is defined to mean the limit of the sum of the first n terms

{displaystyle s_{n}={frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots +{frac {1}{2^{n-1}}}+{frac {1}{2^{n}}}}

as n approaches infinity.

Multiplying s_n by 2 reveals a useful relationship:

{displaystyle 2s_{n}={frac {2}{2}}+{frac {2}{4}}+{frac {2}{8}}+{frac {2}{16}}+cdots +{frac {2}{2^{n}}}=1+left[{frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+cdots +{frac {1}{2^{n-1}}}right]=1+left[s_{n}-{frac {1}{2^{n}}}right].}

Subtracting s_n from both sides,

{displaystyle s_{n}=1-{frac {1}{2^{n}}}.}

As n approaches infinity, s_ntends to 1.

History[edit]

Zeno's paradox[edit]

This series was used as a representation of many of Zeno's paradoxes, one of which, Achilles and the Tortoise, is shown here.^[1] In the paradox, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but by then, the tortoise would already have moved another five meters. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on Zeno argued that the tortoise would always remain ahead of Achilles.

The Eye of Horus[edit]

The parts of the Eye of Horus were once thought to represent the first six summands of the series.^[2]

References[edit]

^Wachsmuth, Bet G. 'Description of Zeno's paradoxes'. Archived from the original on 2014-12-31. Retrieved 2014-12-29.
^Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN978 1 84668 292 6.

Next1.com

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