# How would you solve this (1 Viewer)

#### cossine

##### Well-Known Member
So you would end up with geometric series = num_minutes_in_a_year, where n= num_times_finger_snapped. There is no way to solve this numerically so you will need to use inspection

#### jimmysmith560

##### Le Phénix Trilingue
Moderator
The official solution is as follows:

This question also requires a knowledge of logarithms. Don’t stress if you don’t understand it yet!

To solve a problem like this, it is often helpful to draw a diagram and/or table to illustrate what is happening. It then becomes easier to spot a pattern. We will draw a table to compare the number of finger snaps (n) with the total time elapsed in minutes (t):

Can we find some mathematical expression that will give us the time elapsed for a certain number of finger snaps? Since the time elapsed between finger snaps always doubles, we would expect powers of 2 to be involved in the expression. In fact, we notice that the numbers in the time elapsed column are all 1 less than a power of 2. We could try the expression

$\bg_white t=2^n-1$
but this doesn’t quite work. With a little bit of thought, you should realise that

$\bg_white t=2^{n-1}-1$
is the correct expression. Since we can now calculate the time elapsed for any given number of finger snaps, we can work out how many finger snaps fit into a year. We do this by first finding the number of minutes in a year.

We could of course look this up on the internet; but we really should be able to work it out for ourselves! One year is 365 days, which is 365 × 24 hours, i.e. 8760 hours. This in turn is 8760 × 60 minutes; so there are 525600 minutes in a year.

Now we ask the question: for which value of n (for how many finger snaps) do we get a value of t that is close to the number of minutes in a year? In other words, we want to solve
$\bg_white 525600=2^{n-1}-1$, which gives

$\bg_white 2^{n-1}=525601$

$\bg_white n-1=\log _2\left(525601\right)$

$\bg_white n=\frac{\log _{10}\left(525601\right)}{\:\log _{10}\left(2\right)}+1$

$\bg_white n=20.0036...$

Since we can only have a whole number of finger snaps, we have shown that by following the pattern, we would only snap our fingers 20 times in one year.

I hope this helps!