from height of 8 meters a ball fell down and each time it bounces half the distance back.what will be the distance travelled?
We use the formula for a sum to infinty:
S=a/(1-r) where r is the constant multiple, 0.5 and a is the starting height 8
Therefore S=8/(1-0.5)=16m total travelled
This is the same as adding 8+8/2+8/4+8/8+8/16+8/32...8/2^n
which is 8+4+2+1+0.5+0.25+0.125...which eventually reaches very close to our answer of 16.
ok lets write out the first few terms of the distance it travels to see if we can see a pattern
first it falls 8 then bounces back up to 4 then falls back down 4 the bounces back up 2 and falls back down 2 then bounces back up 1 and falls back down 1.......
we can write this as
$\displaystyle
D=8+4+4+2+2+1+1+\ldots$
grouping the the pairs of terms we have
$\displaystyle
D=8+8+4+2+\ldots$
you should notice that from the second term onwards forms a geometric progression with first term 8 and common ratio 0.5 so we can write it as
$\displaystyle
D=8+\sum_{i=1}^\infty 8(0.5)^{i-1}$
you should know that for common ratio r, first term a and |r|<1 this infinite sum has a value of $\displaystyle \frac{a}{1-r}$
so
$\displaystyle
D=8+\frac{8}{1-0.5}=24m$