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i need mathematical approach to arithmetic progression and geometric progression.

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=Mathematical Designs and patterns can be made using notions of Arithmetic progression and geometric progression. AP techniques can be applied in engineering which helps this field to a large extent....=

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The question cannot be answered because it assumes something which is simply not true. There are some situations in which arithmetic progression is more appropriate and others in which geometric progression is more appropriate. Neither of them is "preferred".

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The common ratio is the ratio of the nth term (n > 1) to the (n-1)th term. For the progression to be geometric, this ratio must be a non-zero constant.

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Immediately springing to mind, geometric progression is used in accountancy in finding the Net Present Value of projects (specifically, the value of money each year based on the discount factor). It is also used in annuities, working out monthly repayments of loans and values of investments - compound interest is a geometric progression.

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It is a geometric progression.

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Arithmetic progression and geometric progression are used in mathematical designs and patterns and also used in all engineering projects involving designs.

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· geometric progression

· geometry

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Yes, the common ratio in a geometric progression can be 1. In a geometric progression, each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. When the common ratio is 1, each term is equal to the previous term, resulting in a sequence of repeated values. This is known as a constant or degenerate geometric progression.

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The geometric series is, itself, a sum of a geometric progression. The sum of an infinite geometric sequence exists if the common ratio has an absolute value which is less than 1, and not if it is 1 or greater.

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the population will exhaust the food supply

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Geology,

Geography,

Geometry,

Gems,

Gold,

Gadolinium,

Gallium,

Germanium,

Graduated Cylinder,

Gametes,

Gauges,

Geotropism,

Gigabytes,

Gigapascal,

Gluon,

and

Gravity.

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Geometric progression 1, 4, 16, 64, 256 would seem to fit...

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Divide any term, except the first, by the term before it.

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Its an arithmetic progression with a step of +4.

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This is a geometric progression with a factor of -10, so 0.562.

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In order to tie a geometric progression to a linear progression. For example, it is easier to calculate 12 log 1.024 than 1.024^12. Exponentials can be simplified.

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In an arithmetic progression the difference between each term (except the first) and the one before is a constant. In a geometric progression, their ratio is a constant. That is, Arithmetic progression U(n) - U(n-1) = d, where d, the common difference, is a constant and n = 2, 3, 4, ... Equivalently, U(n) = U(n-1) + d = U(1) + (n-1)*d Geometric progression U(n) / U(n-1) = r, where r, the common ratio is a non-zero constant and n = 2, 3, 4, ... Equivalently, U(n) = U(n-1)*r = U(1)*r^(n-1).

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This is referred to as a geometric progression - as opposed to an arithmetic progression, where each new number is achieved via addition or subtraction.

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A geometric sequence is : a•r^n while a quadratic sequence is a• n^2 + b•n + c

So the answer is no, unless we are talking about an infinite sequence of zeros which strictly speaking is both a geometric and a quadratic sequence.

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It is a geometric progression with common ratio 0.5

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the population will exhaust the food supply
the population will exhaust the food supply

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Even numbers.

A geometric progression where each number is three times the previous number.

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For an Arithmetic Progression,

Sum = 15[a + 7d].{a = first term and d = common difference}

For a Geometric Progression,

Sum = a[1-r^15]/(r-1).{r = common ratio }.

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No. binary fission in bacteria is a geometric progression. A single bacteria splits into 2 bacteria. They split into 4. They split into 8, then 16 32 64 128 256 512 2048 4096...

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15. It's a Geometric Progression with a Common Ratio of 1/5 (or 0.2).

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Sounds like a Geometric Progression eg 1-3-9-27-81 etc

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It is not possible to answer this question without information on whether the terms are of an arithmetic or geometric (or other) progression, and what the starting term is.

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Well, the factor relevant to this is that probably you have to do mental imaging as well and in the olden days maths could also be applied to principles in various fields like modern physics and hence it was a gray zone of math and was appropriately called geometric since it had its natural implementation in other fields

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There are different answers depending upon whether the sequence is an arithmetic progression, a geometric progression, or some other sequence.

For example, the sequence 4/1 - 4/3 + 4/5 - 4/7 adds to pi

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It is not possible to explain because you have not specified the nature of the sequence.

A sequence can be an arithmetic, or geometric progression, increasing or decreasing. Or it can be a polynomial or power progression, again increasing or decreasing. Or it can be a sequence of random numbers.

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It is neither.

(-6) - (-2) = -4

(-18) - (-6) = -12 which is not the same as -4.

Therefore it is not an arithmetic progression - which requires the difference between successive terms to be the same.

Also

-162/-54 = 3

-468/-162 = 2.88... recurring, and that is not the same as 3.

Therefore it is not a geometric progression - which requires the ratio of terms to be the same.

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Compound interest, radioactive decay.

There are other applications where GP applies for a while - until it reaches a barrier: depreciation, population growth or decline.

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It's a geometric progression with the initial term 1/2 and common ratio 1/2. The infinite sum of the series is 1.

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t(1) = a = 54

t(4) = a*r^3 = 2

t(4)/t(1) = r^3 = 2/54 = 1/27 and so r = 1/3

Then sum to infinity = a/(1 - r) = 54/(1 - 1/3) = 54/(2/3) = 81.

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It appears to have been Svante Arrhenius (1859-1927) in 1896, a Swedish scientist who developed what is now know as the 'greenhouse gas law':

"if the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression"

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In an arithmetic progression the difference between each term (except the first) and the one before is a constant. In a geometric progression, their ratio is a constant.

That is,

Arithmetic progression

U(n) - U(n-1) = d, where d, the common difference, is a constant and n = 2, 3, 4, ...

Equivalently,

U(n) = U(n-1) + d = U(1) + (n-1)*d


Geometric progression

U(n) / U(n-1) = r, where r, the common ratio is a non-zero constant and n = 2, 3, 4, ...

Equivalently,

U(n) = U(n-1)*r = U(1)*r^(n-1).

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In an arithmetic progression (AP), each term is obtained by adding a constant value to the previous term. In a geometric progression (GP), each term is obtained by multiplying the previous term by a constant value. An AP will have a common difference between consecutive terms, while a GP will have a common ratio between consecutive terms.

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In a geometric setting, the inscribed mean (IMA) is always less than or equal to the circumscribed mean (AMA) due to the inequality in a geometric progression ((a \geq g \geq h)). However, in other contexts or disciplines, this relationship may not always hold true.

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The sum to infinity of a geometric series is given by the formula S∞=a1/(1-r), where a1 is the first term in the series and r is found by dividing any term by the term immediately before it.

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Your question is ill-posed. Is there a particular formula (e.g., \sum_{i=0}^{n-1} a r^i = a(1-r^n)/(1-r)) that you're trying to prove?

If so, this page may be some help:

http://www.mathalino.com/reviewer/derivation-of-formulas/sum-of-finite-and-infinite-geometric-progression

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For Geometric Progression

#1 = a = 5

#2 = ar = x

#3 = ar^2 = y

#4 = ar^3 = 40

We need to find 'r'

To do this ,divide #4 by #1 , hence

ar^3 / a = 40 / 5

Hence r^3 = 8 (Notice the 'a' cancels down to leave 'r^3'

Cube root both sides

Hence r = 2

When r = 2

#2 = ar = 5 X 2 = 10 = x

#3 = ar%2 - 5 x 2^2 = 5 x 4 = 20 = y

So the geometric progression is 5,x,y,40 = 5,10,20,40

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Growth whose rate becomes ever more rapid in proportion to the growing total number or size

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression).

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I am pleased with your progression.

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a basic harmonic progression (APEX)

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it is the progression of hominid species.

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