Question

Suppose four distinct positive numbers $a_1,a_2,a_3,a_4$ are in $G.P$. Let $b_1=a_1,b_2=b_1+a_2,b_3=b_2+a_3$ and $b_4=b_3+a_4$.
STATEMENT-1 : The numbers $b_1,b_2,b_3,b_4$ are neither in $A.P$. nor in $G.P$.
STATEMENT-2 : The numbers $b_1,b_2,b_3,b_4$ are in $H.P$.

• A. STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is a correct explanation for STATEMENT1
• B. STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is NOT a correct explanation for STATEMENT1.
• C. STATEMENT1 is True, STATEMENT2 is False
• D. STATEMENT1 is False, STATEMENT2 is True

Answer 1

Answer: (C)

STATEMENT1 is True, STATEMENT2 is False

$a1=k,a2=kr,a3=k{r}^2,a4=k{r}^3$
$b1=k,b2=k+kr,b3=k+kr+k{r}^2,b4=k+kr+k{r}^2+k{r}^3$
Thus, it is obvious that $b1,b2,b3,b4$ are neither in AP nor in GP.
Again, $\frac{1}{b1}+\frac{1}{b3}=\frac{1}{k}+\frac{1}{k+kr+k{r}^2}\ne \frac{2}{k+kr}=\frac{2}{b2}$
Hence, $b1,b2,b3$ are not in HP.
Hence, (C) is correct.