The radii 50, 100, 150, ... form an AP with first term $a=50$ and common difference $d=50$.

<b>(i) Radius of the 13th spiral</b>
Step 1: Use $a_{n}=a+(n-1)d$ with $n=13$.
$$a_{13}=50+(13-1)\times 50=50+600$$
Step 2: Therefore $a_{13}=650$ cm.

<b>(ii) Value of n when the radius is 500 cm</b>
Step 3: Put $a_{n}=500$.
$$50+(n-1)50=500$$
Step 4: Simplify.
$$(n-1)50=450 \;\Rightarrow\; n-1=9 \;\Rightarrow\; n=10$$
So the $10^{\text{th}}$ spiral has radius 500 cm.

<b>(iii)(a) Total saplings till the 11th spiral</b>
Step 5: The numbers of saplings 10, 20, 30, ... form an AP with $a=10$, $d=10$.
Step 6: Use the sum formula with $n=11$.
$$S_{n}=\frac{n}{2}\left[2a+(n-1)d\right]$$
$$S_{11}=\frac{11}{2}\left[2(10)+(11-1)(10)\right]$$
Step 7: Simplify.
$$S_{11}=\frac{11}{2}\left[20+100\right]=\frac{11}{2}\times 120=660$$
So $660$ saplings were planted till the $11^{\text{th}}$ spiral.

<b>OR (iii)(b) Till which spiral are there 450 saplings</b>
Step 8: Put $S_{n}=450$.
$$\frac{n}{2}\left[2(10)+(n-1)(10)\right]=450$$
Step 9: Simplify inside the bracket.
$$\frac{n}{2}\left[10n+10\right]=450 \;\Rightarrow\; 5n(n+1)=450$$
Step 10: Divide by 5.
$$n^{2}+n-90=0$$
Step 11: Factorise ($10\times(-9)=-90$, $10-9=1$).
$$(n+10)(n-9)=0 \;\Rightarrow\; n=9 \text{ or } n=-10$$
Step 12: Since $n$ must be a positive integer, $n=9$.
Hence there will be a total of 450 saplings till the $9^{\text{th}}$ spiral.