Question

Assertion :There are infinite geometric progressions for which 27, 8 and 12 are three of its terms (not necessarily consecutive). Reason: Given terms are integers.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Let, if possible, 8 be the first term and 12 and 27 be $m^{th}$ and $n^{th}$ terms, respectively, then,

$12=a{r}^{m-1}=8r^{m-1},27=8r^{n-1}$

$\Rightarrow \frac{3}{2}=r^{m-1},(\frac{3}{2}{)}^3=r^{n-1}=r^{3(m-1)}$

$\to n-1=3m-3or3m=n+2$

$\Rightarrow \frac{m}{1}=\frac{n+2}{3}=k$ (say)

$\therefore m=k,n=3k-2$

By giving $k$ different values, we get the integral values of $m$ and $n$. Hence, there can be infinite number of G.P.'s whose any three terms will be $8,12,27$ (not consecutive). Obviously, statement 2 is not a correct expalination of statement 1.