If 2,root 8,4,.......in g.p,then the common ratio is...with procedure
Answers
Answer:
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Step-by-step explanation:
In mathematics, a geometric progression(sequence) (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.
The geometric progression can be written as:
ar0=a, ar1=ar, ar2, ar3, ...
where r ≠ 0, r is the common ratio and a is a scale factor(also the first term).
Examples
A geometric progression with common ratio 2 and scale factor 1 is
1, 2, 4, 8, 16, 32...
A geometric sequence with common ratio 3 and scale factor 4 is
4, 12, 36, 108, 324...
A geometric progression with common ratio -1 and scale factor 5 is
5, -5, 5, -5, 5, -5,...
Formulas
Formula for the n-th term can be defined as:
an = an-1⋅r
an = a1⋅rn-1
Formula for the common ratio is:
r =
ak
ak-1
If the common ratio is:
Negative, the results will alternate between positive and negative.
Example:
1, -2, 4, -8, 16, -32... - the common ratio is -2 and the first term is 1.
Greater than 1, there will be exponential growth towards infinity (positive).
Example:
1, 5, 25, 125, 625 ... - the common ratio is 5.
Less than -1, there will be exponential growth towards infinity (positive and negative).
Example:
1, -5, 25, -125, 625, -3125, 15625, -78125, 390625, -1953125 ... - the common ratio is -5.
Between 1 and -1, there will be exponential decay towards zero.
Example:
4, 2, 1, 0.5, 0.25, 0.125, 0.0625 ... - the common ratio is \displaystyle \frac{1}{2}
2
1
4, -2, 1, -0.5, 0.25, -0.125, 0.0625 ... - the common ratio is \displaystyle -\frac{1}{2}−
2
1
.
Zero, the results will remain at zero.
Example:
4, 0, 0, 0, 0 ... - the common ratio is 0 and the first term is 4.
Geometric Progression Properties
a2k = ak-1⋅ak+1
a1⋅an = a2⋅an-1 =...= ak⋅an-k+1
Formula for the sum of the first n numbers of a geometric series
Sn =
a1 - anr
1 - r
= a1.
1 - rn
1 - r
Infinite geometric series where |r| < 1
If |r| < 1 then an -> 0, when n -> ∞.
The sum S of such an infinite geometric series is given by the formula:
S = a1
1
1 - r
which is valid only when |r| < 1.
a1 is the first term.
given 2,√8,4 are in gp then r=(√8)/2=√2