check whether the following sequence are in gp 1/2,1,2,4
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Answer:
YES it is in sequence
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In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Diagram illustrating three basic geometric sequences of the pattern 1(rn−1) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
Examples of a geometric sequence are powers rk of a fixed number r, such as 2k and 3k. The general form of a geometric sequence is
{\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots
where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Difference here means the second minus the first. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2.
Visual proof of the derivation of arithmetic progression formulas – the faded blocks are a rotated copy of the arithmetic progression
If the initial term of an arithmetic progression is {\displaystyle a_{1}}a_{1} and the common difference of successive members is d, then the nth term of the sequence ({\displaystyle a_{n}}a_{n}) is given by:
{\displaystyle \ a_{n}=a_{1}+(n-1)d}{\displaystyle \ a_{n}=a_{1}+(n-1)d},
and in general
{\displaystyle \ a_{n}=a_{m}+(n-m)d}{\displaystyle \ a_{n}=a_{m}+(n-m)d}.
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
positive, then the members (terms) will grow towards positive infinity;
negative, then the members (terms) will grow towards negative infinity.
In mathematics, an arithmetico–geometric sequence is the result of the term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put more plainly, the nth term of an arithmetico–geometric sequence is the product of the nth term of an arithmetic sequence and the nth term of a geometric one. Arithmetico–geometric sequences arise in various applications, such as the computation of expected values in probability theory. For instance, the sequence
{\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{16}}},\ {\dfrac {\color {blue}{5}}{\color {green}{32}}},\cdots }{\displaystyle {\dfrac {\color {blue}{0}}{\color {green}{1}}},\ {\dfrac {\color {blue}{1}}{\color {green}{2}}},\ {\dfrac {\color {blue}{2}}{\color {green}{4}}},\ {\dfrac {\color {blue}{3}}{\color {green}{8}}},\ {\dfrac {\color {blue}{4}}{\color {green}{16}}},\ {\dfrac {\color {blue}{5}}{\color {green}{32}}},\cdots }
is an arithmetico–geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green).
The summation of this infinite sequence is known as a arithmetico–geometric series, and its most basic form has been called Gabriel's staircase:[1][2][3]
{\displaystyle \sum _{k=1}^{\infty }{\color {blue}k}{\color {green}r^{k}}={\frac {r}{(1-r)^{2}}},\quad \mathrm {for\ } 0<r<1}{\displaystyle \sum _{k=1}^{\infty }{\color {blue}k}{\color {green}r^{k}}={\frac {r}{(1-r)^{2}}},\quad \mathrm {for\ } 0<r<1}
The denomination may also be applied to different objects presenting characteristics of both arithmetic and geometric sequences; for instance the French notion of arithmetico–geometric sequence refers to sequences of the form {\displaystyle u_{n+1}=au_{n}+b}{\displaystyle u_{n+1}=au_{n}+b}, which generalise both arithmetic and geometric sequences. Such sequences are a special case of linear difference equations.
