Sequences & Series
AP, GP and logarithmic relations
nta_pyq_2023_jan
Grade 11
Question:
For three positive integers p, q, r, $x^{pq^2} = y^{qr} = z^{p^2r}$ and $r = pq + 1$ such that $3, 3\log_y x, 3\log_z y, 7\log_x z$ are in A.P. with common difference $\dfrac{1}{2}$. Then $r - p - q$ is equal to
Step-by-Step Solution
Key Concept: Let the common value be $\lambda$. Then $pq^2 = \log_x \lambda$, $qr = \log_y \lambda$, $p^2r = \log_z \lambda$. Express $\log_y x$, $\log_z y$, $\log_x z$ in terms of p, q, r, then use the AP condition.
From the AP: $\frac{3r}{pq} - 3 = \frac{1}{2} \Rightarrow r = \frac{7}{6}pq$. With $r = pq+1$: $pq = 6$, $r = 7$. Also $\frac{3p^2}{q} = 3 + 1 = 4 \Rightarrow p^2 = \frac{4q}{3}$. From $pq=6$: $p=2, q=3$. So $r-p-q = 7-2-3 = 2$.
Correct Answer: 1