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Q: Arithmetic and geometric sequences

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how are arithmetic and geometric sequences similar

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Arithmetic : (First term)(last term)(act of terms)/2 Geometric : (first term)(total terms)+common ratio to the power of (1+2+...+(total terms-1))

1.The Geometric mean is less then the arithmetic mean. GEOMETRIC MEAN < ARITHMETIC MEAN 2.

The difference between arithmetic and geometric mean you can find in the following link: "Calculation of the geometric mean of two numbers".

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how are arithmetic and geometric sequences similar

Exponentail functions

an arithmetic sequeunce does not have the sum to infinty, and a geometric sequence has.

An arithmetic-geometric mean is a mean of two numbers which is the common limit of a pair of sequences, whose terms are defined by taking the arithmetic and geometric means of the previous pair of terms.

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Arithmetic : (First term)(last term)(act of terms)/2 Geometric : (first term)(total terms)+common ratio to the power of (1+2+...+(total terms-1))

There is no simple answer. There are simple formulae for simple sequences such as arithmetic or geometric progressions; there are less simple solutions arising from Taylor or Maclaurin series. But for the majority of sequences there are no solutions.

A few examples: Counting numbers are an arithmetic sequence. Radioactive decay, (uncontrolled) bacterial growth follow geometric sequences. The Fibonacci sequence is widespread in nature.

Geometric

1.The Geometric mean is less then the arithmetic mean. GEOMETRIC MEAN < ARITHMETIC MEAN 2.

they look like arithmetic and geometric patterns in math

There are different types of sequences such as arithmetic sequences, geometric sequences, and Fibonacci sequences. Sequences are used in mathematics to study patterns, predict future terms, and model real-world situations, such as population growth or financial investments. Patterns in sequences can help in making predictions and solving problems in various fields like engineering, physics, and computer science.