Question

Which one of the following is the correct statement if ${\mathrm{Q}}_{\mathrm{r}}$ is the difference between two consecutive terms of a progression whose general term is ${\mathrm{T}}_{\mathrm{r}}=3{\mathrm{r}}^2+2\mathrm{r}–1$?

• A. ${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,......$are in $\mathrm{A}.\mathrm{P}.$ with common difference $10$
• B. ${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,......$are in $\mathrm{A}.\mathrm{P}.$ with common difference $6$
• C. ${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,......$are in $\mathrm{A}.\mathrm{P}.$ with common difference $12$
• D. ${\mathrm{Q}}_1={\mathrm{Q}}_2={\mathrm{Q}}_3=.....$

Answer 1

The correct option is B

${\mathrm{Q}}_1,{\mathrm{Q}}_2,{\mathrm{Q}}_3,......$are in $\mathrm{A}.\mathrm{P}.$ with common difference $6$

Explanation for the correct option:

Option (B):

$\because {\mathrm{T}}_{\mathrm{r}}=3{\mathrm{r}}^2+2\mathrm{r}–1$

$\begin{array}{rcl}\therefore {\mathrm{T}}_{(\mathrm{r}+1)}& =& 3{(\mathrm{r}+1)}^2+2(\mathrm{r}+1)–1\\ & =& 3({\mathrm{r}}^2+2\mathrm{r}+1)+2\mathrm{r}+2-1\\ & =& 3{\mathrm{r}}^2+6\mathrm{r}+3+2\mathrm{r}+2-1\\ & =& 3{\mathrm{r}}^2+8\mathrm{r}+4\end{array}$

As it is given that ${\mathrm{Q}}_{\mathrm{r}}$ is the difference between two consecutive terms of a progression whose general term is ${\mathrm{T}}_{\mathrm{r}}=3{\mathrm{r}}^2+2\mathrm{r}–1$.

$$\begin{gathered} {\mathrm{Q}}_{\mathrm{r}}={\mathrm{T}}_{(\mathrm{r}+1)}–{\mathrm{T}}_{\mathrm{r}} \\ {\mathrm{Q}}_{\mathrm{r}}=[3{(\mathrm{r}+1)}^2+2(\mathrm{r}+1)–1]–[3{\mathrm{r}}^2+2\mathrm{r}–1] \\ {\mathrm{Q}}_{\mathrm{r}}=(3{\mathrm{r}}^2+8\mathrm{r}+4)-(3{\mathrm{r}}^2+2\mathrm{r}-1) \\ {\mathrm{Q}}_{\mathrm{r}}=3{\mathrm{r}}^2+8\mathrm{r}+4-3{\mathrm{r}}^2-2\mathrm{r}+1 \\ {\mathrm{Q}}_{\mathrm{r}}=6\mathrm{r}+5 \\ {\mathrm{Q}}_{(\mathrm{r}+1)}=6(\mathrm{r}+1)+5 \end{gathered}$$

Now, the difference between ${\mathrm{Q}}_{\mathrm{r}}$ and ${\mathrm{Q}}_{(\mathrm{r}+1)}$ $={\mathrm{Q}}_{(\mathrm{r}+1)}–{\mathrm{Q}}_{\mathrm{r}}$:

$$\begin{gathered} =\left[6\right(\mathrm{r}+1)+5]–[6\mathrm{r}+5] \\ =6r+6+5-6r-5 \\ =6 \end{gathered}$$

As the difference between ${\mathrm{Q}}_{\mathrm{r}}$ and ${\mathrm{Q}}_{(\mathrm{r}+1)}$ is independent of $r$, so ${\mathrm{Q}}_{\mathrm{r}}$ is the general term of an A.P whose common difference is $6$.

Explanation for the incorrect options:

Option (A), (C) and (D): Since the common difference is calculated as $6$ which is not equal to the value mentioned Option (A), (C) and (D)

Therefore, option (A), (C) and (D) are incorrect.

Hence, Option (B) is the correct option.