A tree, in each year grows 5 cm less than it grew in the previous year. If it grew half a metre in the first year, then the height of the tree (in metres ) when it ceases to grow, is

Let $a_n$ denote the growth of height in the $n^{th}$ year. We have 

$a_1=50 \mathrm{cm} , a_r=50-5(r-1)=55-5r$

The tree ceases to grow in the $r^{th}$ year where $a_r = 0\Rightarrow r=11$

So height of tree is

$\begin{array}{l}{a}_1+a_2+\cdots +a_{10}\\ =\frac{10}{2}[50+5] \mathrm{cm}=275 \mathrm{cm}=2.75 \mathrm{m}\end{array}$