$\text{Let the position vectors of the points}P,Q,R\text{and}S\text{be}\overrightarrow{a}=\hat{i}+2\hat{j}-5\hat{k},$

$\overrightarrow{b}=3\hat{i}+6\hat{j}+3\hat{k},$

$\overrightarrow{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k},\text{and}\overrightarrow{d}=2\hat{i}+\hat{j}+\hat{k},\text{respectively.}$

$\text{Then which of the following statements is true?}$

The correct option is:

$\frac{\overrightarrow{b}+2\overrightarrow{d}}{3}\text{is the position vector of a point which divides}PR\text{internally in the ratio}5:4.$

Step-by-step Breakdown:

Step 1: Write down the given position vectors of points P, Q, R, and S.

$\text{Position vector of}P:\overrightarrow{a}=\hat{i}+2\hat{j}-5\hat{k}$

$\text{Position vector of}Q:\overrightarrow{b}=3\hat{i}+6\hat{j}+3\hat{k}$

$\text{Position vector of}R:\overrightarrow{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}$

$\text{Position vector of}S:\overrightarrow{d}=2\hat{i}+\hat{j}+\hat{k}$

Step 2: Evaluate the expression $\frac{\overrightarrow{b}+2\overrightarrow{d}}{3}$.

First, calculate $2\overrightarrow{d}=2(2\hat{i}+\hat{j}+\hat{k})=4\hat{i}+2\hat{j}+2\hat{k}$.

Now, add $\overrightarrow{b}$ and $2\overrightarrow{d}$:

$\overrightarrow{b}+2\overrightarrow{d}=(3\hat{i}+6\hat{j}+3\hat{k})+(4\hat{i}+2\hat{j}+2\hat{k})=7\hat{i}+8\hat{j}+5\hat{k}$

Dividing by 3 gives:

$\frac{\overrightarrow{b}+2\overrightarrow{d}}{3}=\frac{7}{3}\hat{i}+\frac{8}{3}\hat{j}+\frac{5}{3}\hat{k}$

Step 3: Find the position vector of the point dividing PR internally in the ratio 5 : 4.

By the section formula, the position vector of a point dividing the line joining two points with position vectors $\overrightarrow{a}$ and $\overrightarrow{c}$ internally in the ratio $m:n=5:4$ is given by:

$\overrightarrow{r}=\frac{5\overrightarrow{c}+4\overrightarrow{a}}{5+4}=\frac{5\overrightarrow{c}+4\overrightarrow{a}}{9}$

Calculating $5\overrightarrow{c}$:

$5\overrightarrow{c}=5\left(\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}\right)=17\hat{i}+16\hat{j}+35\hat{k}$

Calculating $4\overrightarrow{a}$:

$4\overrightarrow{a}=4(\hat{i}+2\hat{j}-5\hat{k})=4\hat{i}+8\hat{j}-20\hat{k}$

Summing both vectors:

$5\overrightarrow{c}+4\overrightarrow{a}=(17+4)\hat{i}+(16+8)\hat{j}+(35-20)\hat{k}=21\hat{i}+24\hat{j}+15\hat{k}$

Dividing by 9:

$\overrightarrow{r}=\frac{21\hat{i}+24\hat{j}+15\hat{k}}{9}=\frac{7}{3}\hat{i}+\frac{8}{3}\hat{j}+\frac{5}{3}\hat{k}$

Conclusion:
Since both expressions yield the identical position vector $\frac{7}{3}\hat{i}+\frac{8}{3}\hat{j}+\frac{5}{3}\hat{k}$, the stated option is true.