Question

In a potato race, a bucket is placed at the starting point, which is $5m$ from the first potato, and the other potatoes are placed $3m$ apart in a straight line. There are $n$ potatoes in the line. A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the the bucket to drop it in the bucket, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

Answer 1

We have,

$d_1=$ Distance run by the competitor to pick up first potato $=2\times 5m$

$d_2=$ Distance run by the competitor to pick up second potato $=2(5+3)m$

$d_3=$ Distance run by the competitor to pick up third potato $=2(5+2\times 3)m$

$d_4=$ Distance run by the competitor to pick up first potato $=2(5+3\times 3)m$

. .

. .

. .

$d_n=$ Distance run by the competitor to pick up $n^{th}$ potato $=2[5+(n-1)\times 3]m$

$\therefore$ Total distance run by the competitor to pick up $n$ potatoes

$=d_1+d_2+d_3+....+d_n$

$=2\times 5+2(5+3)+2(5+2\times 3)+2(5+3\times 3)+....+2\{5+(n-1)\times 3\}$ metres

$=2[5+\{5+3\}+\{5+(2\times 3)\}+\{5-(3\times 3)\}+....+\{5+(n-1)\times 3\left\}\right]$

$=2[\underset{n-times}{(5+5+....+5)}+\{3+(2\times 3)+(3\times 3)+....+(n-1)\times 3\left\}\right]$

$=2[5n+3\{1+2+3+....+(n-1)\left\}\right]$

$=2[5n+3\left(\frac{n-1}{2}\right)\{1+(n-1)\left\}\right]$ $[Using{S}_n=\frac{n}{2}(a+1\left)\right]$

$=2[5n+\frac{3n(n-1)}{2}]$

$=[10n+3n(n-1\left)\right]=3n^2+7n=n(3n+7)$ metres

Total distance covered is $n(3n+7)$ meters.