A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step.

(i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?

(ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?

(iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following; (a) − 3 + 2 − … = − 8 (b) 4 − 2 + … = 8. In (a) the sum (− 8) represents going down by eight steps. So, what will the sum 8 in (b) represent?

Let the steps moved down be represented by positive integers and the steps moved up be represented by negative integers.

(i) Initially, the monkey was at step = 1

After 1^{st} jump, the monkey will be at step = 1 + 3 = 4

After 2^{nd} jump, the monkey will be at step = 4 + (−2)
= 2

After 3^{rd} jump, the monkey will be at step = 2 + 3 = 5

After 4^{th} jump, the monkey will be at step = 5 + (−2)
= 3

After 5^{th} jump, the monkey will be at step = 3 + 3 = 6

After 6^{th} jump, the monkey will be at step = 6 + (−2)
= 4

After 7^{th} jump, the monkey will be at step = 4 + 3 = 7

After 8^{th} jump, the monkey will be at step = 7 + (−2)
= 5

After 9^{th} jump, the monkey will be at step = 5 + 3 = 8

After 10^{th} jump, the monkey will be at step = 8 + (−2)
= 6

After 11^{th} jump, the monkey will be at step = 6 + 3 = 9

Clearly, the monkey will be at water level (i.e., 9^{th}
step) after 11 jumps.

(ii) Initially, the monkey was at step = 9

After 1^{st} jump, the monkey will be at step = 9 + (−4)
= 5

After 2^{nd} jump, the monkey will be at step = 5 + 2 = 7

After 3^{rd} jump, the monkey will be at step = 7 + (−
4) = 3

After 4^{th} jump, the monkey will be at step = 3 + 2 = 5

After 5^{th} jump, the monkey will be at step = 5 + (−
4) = 1

Clearly, the monkey will reach back at the top step after 5 jumps.

(iii) If steps moved down are represented by negative integers and steps moved up are represented by positive integers, then his moves will be as follows.

__Moves in part (i)__

− 3 + 2 − 3 + 2 − 3 + 2 − 3 + 2 − 3 + 2 − 3 = −8

__Moves in part (ii)__

4 − 2 + 4 − 2 + 4 = 8

Moves in part (ii) represent going up 8 steps.

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