Question

If a clock strikes once at $1o^{\prime }clock$, twice at $2o^{\prime }clock$ and so on, how many times will it strike in a day?

Answer 1

There are $24$ hours in a day and clock has $12$ hours, therefore, there will be two rounds of clock per day. Let us write this as an arithmetic series because we have to find the number of times that the clock strike in a day. Thus, the A.P is

$1,2,3,4,....,12$

We know that the sum of an arithmetic series with first term $a$ and common difference $d$ is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Now to find the sum of series, substitute $n=12,a=1$ and $d=2-1=1$ in $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ as follows:

$S_{12}=\frac{12}{2}\left[\right(2\times 1)+(12-1\left)1\right]=6[2+(11\times 1\left)\right]=6(2+11)=6\times 13=78$

Therefore, it will strike $78$ times in $12$ hours.

Now to find the number of times it will strike in full day that is in $24$ hours is $2\times 78=156$.

Hence, the clock strikes $156$ times in a day.