Question

If \[f(x) = {2^x}\], then find what \[f(0),f(1),f(2)...\] follows____________
A) A.P.
B) G.P.
C) H.P.
D) A.G.P\[x\]

Answer 1

Hint: To find the type of the series, at first, we will find the value of the function at the given points. Then by the form of the series we will identify the type of the series.

Complete step by step answer:
It is given that,
\[f(x) = {2^x}\]
At first, we will find the value of \[f(0),f(1),f(2)\] by substituting the values of .
Let us substitute \[x = 0\] in the given function \[f(x) = {2^x}\] we get,
\[f(0) = {2^0} = 1\]
Let us substitute \[x = 1\] in the given function \[f(x) = {2^x}\] we get,
\[f(1) = {2^1} = 2\]
Let us substitute \[x = 2\] in the given function \[f(x) = {2^x}\] we get,
\[f(2) = {2^2} = 4\]
The series can be written as \[1,2,4...\]
Now we have to find the common ratio of the series,
Common ratio of the geometric series is $d = \dfrac{{a(n)}}{{a(n - 1)}}$
Where $a(n)$ is the term of the series and $a(n - 1)$ is the previous term of the nth term.
Let n=3 that is $a(n) = 4$ and $a(n - 1) = 2$
$d = \dfrac{4}{2} = 2$
Hence the common ratio is 2.
This series is in G.P. whose initial term is \[1\] and the common ratio is \[2.\]

Hence, the series \[f(0),f(1),f(2)...\] are in G.P.

Note:
 A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same.
The general form of A.P. is \[a,(a + d),(a + 2d),...,[a + (n - 1)d]\] Where, \[a\]is the initial term and \[d\] be the common difference.
A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is always the same.
The general form of G.P. is \[a,ar,a{r^2},...,a{r^n}\]. Where, \[a\]is the initial term and \[r\] be the common ratio.
A sequence of numbers is called a harmonic progression if the reciprocal of the terms is in AP.
A series is said to be an arithmetic-geometric series if each term is formed by multiplying the corresponding term of an AP and a GP.
The general form of A.G.P. is . Where, is the initial term and be the common difference and be the common ratio.
\[a,(a + d)r,(a + 2d){r^2},...,[a + (n - 1)d]{r^{n - 1}}\]