(Q37).In a bucket and ball race, a bucket is placed at the starting point, which is 5 m from the first ball, and the other balls are placed 3 m apart in a straight line. There are ten balls in the line.A competitor starts from the bucket, picks up the nearest ball, runs back with it, drops it in the bucket, runs back to pick up the next ball, runs to the bucket to drop it in, and she continues in the same way until all the balls are in the bucket. What is the total distance the competitor has to run?

[Hint: To pick up the first ball and the second ball, the total distance (in metres) run by a competitor is 2 × 5 + 2 × (5 + 3)]

Given: Balls are placed at an equal distance of 3 m from one another.

Distance of first ball from the bucket = 5 m

Distance of second ball from the bucket = + 3 = m ( + 1 × 3)

Distance of third ball from the bucket = + 3 = m ( + 2 × 3)

Distance of fourth ball from the bucket = + 3 = m ( + 3 × 3)

...................

∴ Distance of the tenth ball from the bucket = + 9 × = + = m.

1st ball: Distance covered by the competitor in picking up and dropping it in the bucket = 2 × = m.

2nd ball: Distance covered by the competitor in picking up and dropping it in the bucket = 2 × = m.

3rd ball: Distance covered by the competitor in picking up and dropping it in the bucket = 2 × = m.

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10th ball: Distance covered by the competitor in picking up and dropping it in the bucket = 2 × = m.

Total distance = 10 m + 16 m + 22 m + …… + 64 m.

Clearly, this is an A.P in which a = ; d = a2 – a1 = – = and n = .

∴ Sn =
n
2
 [2a + (n-1)d]
S10 =
2
 [2× + (-1)]

= [ + ]

= ×

= m

∴ Total distance = m.