Question

If a sequence or series is not directly in $\text{A.P}$ or $\text{G.P}$, then their general terms $\left(l_n\right)$ cannot be determined easily. For such cases to determine $t_n$ we use the following steps.
Step 1: First of all we find the successive difference (first difference, second difference, third difference $\dots$ so on).
Step 2: If first successive difference is in $\text{A.P}$, then general term can be taken as $t_n=a{n}^2+bn+c$ (i.e. consider $t_n$ as quadratic polynomial in the decreasing power of $n$ with constants $a,b,c$ ).
Step 3: If the second, third successive differences are in $\text{A.P}$, then $t_n=a{n}^3+b{n}^2+cd+d$ for second successive difference in $\text{A.P}$ and so on where $a,b,c,d$ are constants whose order can be changed.
Step 4: If the first difference in step 1 is in $\text{G.P}$, then take $t_n=a{r}^{n-1}+bn+c$. Similarly if second, third difference are in $\text{G .P}$, then general terms are considered by $t_n=a{r}^{n-1}+b{n}^2+cn+d$ and $t_n=a{r}^{n-1}+b{n}^3+c{n}^2+dn+e$ respectively where $a,b,c,d,e\dots$ are constants whose orders can be changed.
Now consider the sequences
$P:9,16,29,54,103,\dots$
$Q:4,14,30,52,80,114,\dots$
$R:2,12,36,80,150,252,\dots$
$S:2,5,12,31,86,\dots$
On the basis of above data (information) answer the following questions:If $n^{th}$ term of sequence $P$ & $R$ are respectively of the forms $a{2}^{n-1}+bn+c$ & $A+Bn+C{n}^2+D{n}^3$, then the value of $(a+b+c)(A+B+C+D)$ equals

• A. $0$
• B. $18$
• C. $9$
• D. $12$

Answer 1

The correct option is B $18$

Given that, $P:9,16,29,54,103,...$
and $R:2,12,36,80,150,252,...$
$t_n$ for sequence $P$ is $a{2}^{n-1}+bn+c$
$i.e{t}_n=a{2}^{n-1}+bn+c$
Putting $n=1$ for first term
$\therefore a+b+c=t_1=9$
and $t_n$ for $R$ is given by $t_n=A+Bn+C{n}^2+D{n}^2$
Putting $n=1$ for the first term we get
$\therefore A+B+C+D=t_1=2$
Therefore, $(a+b+c)(A+B+C+D=2)=9\times 2=18$
Ans: B