In a potato race a bucket is placed at the starting point. It is $5 m$ away from the first potato. The rest of the potatoes are placed in a straight line each $3 m$ away from the other. Each competitor starts from the bucket. Picks up the nearest potato and runs back and drops it in the bucket and continues till all potatoes are placed in the bucket. What is the total distance covered if $15$ potatoes are placed in the race?
If the distance covered is $1340 m$ , find the number of potatoes?

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Solution

Given,

1st potato distance $= 5 \times 2 = 10$

2nd potato distance $= (5 + 3) \times 2 = 16$

3rd potato distance $= (8 + 3) \times 2 = 22$

and so on.

we get the A.P. series $10, 16, 22, . . . . . . . . . .$

with $a = 10, d = 6$

Formula,

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$= \frac{15}{2} [2 (10) + (15 - 1) 6]$

$= \frac{15}{2} [104]$

$= 780$

Therefore the distance covered when 15 potatoes are placed $= 780 m$

Given,

$S_{n} = 1340$

$S_{n} = \frac{n}{2} [2 (10) + (n - 1) 6]$

$1340 = \frac{n}{2} [14 + 6 n]$

$6 n^{2} + 14 n - 2680 = 0$

solving the above quadratic equation, we get,

$n = 20, - \frac{67}{3}$

Therefore the number of potatoes are $20$ .

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