Question

If a sequence or series is not a direct form of an A.P, G.P, etc., then its ${n^{th}}$ term cannot be determined. In such cases, we use the following steps to find the ${n^{th}}$ term of the given sequence.
$Step1$ : Find the difference between the successive terms of the given sequence. If these differences are in A.P, then take ${T_n} = a{n^2} + bn + c$ , where $a,b,c$ are constants.
$Step2$ : If the successive differences finding in $Step1$ are in G.P with common ratio $r$ , then take ${T_n} = a + bn + c{r^{n - 1}}$ , where $a,b,c$ are constants.
$Step3$ : If the second successive differences (Differences of the differences) in $Step1$ are in A.P, then take ${T_n} = a{n^3} + b{n^2} + cn + d$ , where $a,b,c,d$ are constants.
$Step4$ : If the second successive differences (Differences of the differences) in $Step1$ are in G.P, then take ${T_n} = a{n^2} + bn + c + d{r^{n - 1}}$ , where $a,b,c,d$ are constants.
Now let sequences be:
$C:\ln 2,\ln 4,\ln 32,\ln 1024,....$
On the basis of the above information, answer the following question:
The format of ${n^{th}}$ the term ${T_n}$ of the sequence $C$ is:
$\left( a \right){\text{ }}a{n^2} + bn + c$
$\left( b \right){\text{ }}a{n^3} + b{n^2} + cn + d$
$\left( c \right){\text{ an + b + c}}{{\text{r}}^{n - 1}}$
$\left( d \right){\text{ }}a{n^2} + bn + c + d{r^{n - 1}}$

Answer 1

For solving this type of question we should check for the possibilities of any condition satisfying in the above steps given, either we have to find the difference, differences of differences, or common ratio, or then we have to justify our answer with the above options given.
Formula used:
Logarithmic properties used are:
$\ln {a^n} = n\ln a$

Complete step by step solution:
So we have the sequences given as $\ln 2,\ln 4,\ln 32,\ln 1024,....$
So now we will try to convert them into the series in terms of $\ln 2$ the number in the sequence in the form of ${2^n}$ .
Therefore, on doing it we have the sequence as
$\Rightarrow \ln 2,\ln {2^2},\ln {2^5},\ln {2^{10}},...$
Now using the properties of logarithmic, we get the above equation as
$\Rightarrow \ln 2,2\ln 2,5\ln 2,10\ln 2,...$
Now from this, we can see that the difference between them is $1,3,5,7,...$ but the difference of difference is $2$ which satisfies $Step \,3$ .
That is if the second successive difference which will the difference of the differences in $Step1$ are in the arithmetic progression,
Then we take ${T_n} = a{n^3} + b{n^2} + cn + d$ , where $a,b,c,d$ are constants.

Hence, the option $\left( c \right)$ is correct.

Additional information:
Arithmetic progression is a sequence of numbers in which each differs by a constant value. Whereas geometric progression is a sequence of numbers in which the successor is found by multiplying the previous number with a constant number. And harmonic progression is the sequence of numbers in which the reciprocals differ by a constant value.

Note:
For solving the questions like this we should always try to convert it into the common value if the sequence is given in logarithmic. And then we have to check every step if it satisfies arithmetic progression or geometric progression.