For three positive integers p,q,r,x^pq^2 = y^qr = z^p^2r and r = pq + 1 such that 3,3_yx,3_zy,7_xz are in A.P. with common difference 1/2. The r - p - q is equal to

aligned & pq^2= _x \\ & qr= _y \\ & p^2 r= _2 \\ & _y x=qrpq^2=rpq (1) \\ & _x z=pq^2p^2 r=q^2pr . .(2) \\ & _z y=p^2 rqr=p^

For three positive integers p,q,r,x^{pq^{2}} = y^{qr} = z^{p^{2}r} 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 1/2. The r - p - q is equal to: Sequence & Series (Mathematics)

$\begin{aligned} & \mathrm{pq}^2=\log _{\mathrm{x}} \lambda \\ & \mathrm{qr}=\log _{\mathrm{y}} \lambda \\ & \mathrm{p}^2 \mathrm{r}=\log _2 \lambda \\ & \log _{\mathrm{y}} \mathrm{x}=\frac{\mathrm{qr}}{\mathrm{pq}^2}=\frac{\mathrm{r}}{\mathrm{pq}} \ldots \ldots(1) \\ & \log _{\mathrm{x}} \mathrm{z}=\frac{\mathrm{pq}^2}{\mathrm{p}^2 \mathrm{r}}=\frac{\mathrm{q}^2}{\mathrm{pr}} \ldots \ldots . .(2) \\ & \log _{\mathrm{z}} \mathrm{y}=\frac{\mathrm{p}^2 \mathrm{r}}{\mathrm{qr}}=\frac{\mathrm{p}^2}{\mathrm{q}} \ldots \ldots .(3) \\ & 3, \frac{3 \mathrm{r}}{\mathrm{pq}}, \frac{3 \mathrm{p}^2}{\mathrm{q}}, \frac{7 \mathrm{q}^2}{\mathrm{pr}} \text { in A.P } \\ & \frac{3 \mathrm{r}}{\mathrm{pq}}-3=\frac{1}{2} \\ & \mathrm{r}=\frac{7}{6} \mathrm{pq} \ldots \ldots .(4) \\ & \mathrm{r}=\mathrm{pq}+1 \\ & \mathrm{pq}=6 \ldots \ldots \ldots \ldots . .(5) \\ & \mathrm{r}=7 \ldots \ldots \ldots \ldots \ldots . .(6) \\ & \frac{3 \mathrm{p}^2}{\mathrm{q}}=4 \\ & \left(\frac{3 \mathrm{p}^2}{q}\right)(p q)=4 \times 6 \\ & \Rightarrow 3 p^3=\mathbf{2 4} \\ & \Rightarrow p^3=8=\mathbf{2}^3 \\ & \Rightarrow \boldsymbol{p}=\mathbf{2} \\ & \therefore \boldsymbol{q}=\frac{6}{\mathbf{p}}=\frac{6}{\mathbf{2}}=\mathbf{3} \\ & \therefore \boldsymbol{r}-\boldsymbol{p}-\boldsymbol{q}=\mathbf{7}-\mathbf{2}-\mathbf{3}=\mathbf{7}-\mathbf{5}=\mathbf{2}\end{aligned}$

Explanation

Let a_{1},a_{2},,a_{n} be a given A.P. whose common difference is an integer and S_{n} = a_{1} + a_{2} + + a_{n}. If a_{1} = 1,a_{n} = 300 and 15 \leq n \leq 50, then the ordered pair ( S_{n - 4},a_{n - 4} ) is equal to

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