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: -3 + 2 -.....= -8 ; 4 -2 + 4 -2 .... = 8. In (a), the sum (-8) represents going down by eight steps. So, what will the sum 8 in (b) represent?

(i) Let us assume, down moved steps be represented by positive whereas up moved steps be represented by negative integers

It is given that,

Initially the monkey will be at step = 1

After the 1^{st} jump monkey’s step = 1 + 3 = 4

After the 2^{nd} jump monkey’s step = 4 + (-2) = 2

After the 3^{rd} jump monkey’s step = 2 + 3 = 5

After the 4^{th} jump monkey’s step = 5 + (-2) = 3

After the 5^{th} jump monkey’s step = 3 + 3 = 6

After the 6^{th} jump monkey’s step = 6 + (-2) = 4

After the 7^{th} jump monkey’s step = 4 + 3 = 7

After the 8^{th} jump monkey’s step = 7 + (-2) = 5

After the 9^{th} jump monkey’s step = 5 + 3 = 8

After the 10^{th} jump monkey’s step = 8 + (-2) = 6

After the 11^{th} jump monkey’s step = 6 + 3 = 9

Hence,

From above results it is clear that the monkey will be at water level i.e.,9 after 11 jumps

(ii) From above, we have

Initial step of monkey = 9

Therefore,

After the 1^{st} jump monkey’s step = 9 + (-4) = 5

After the 2^{nd} jump monkey’s step = 5 + 2 = 7

After the 3^{rd} jump monkey’s step = 7 + (-4) = 3

After the 4^{th} jump monkey’s step = 3 + 2 = 5

After the 5^{th} jump monkey’s step = 5 + (-4) = 1

Hence,

It is clear from the above results is that after 5 jumps the monkey will reach back at the top

(iii) If we will represent the down steps by negative and up steps by positive than the moves are are as follows:

We have,

Moves in (i):

- 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 = 8

Also,

Moves in part (ii):

4 – 2 + 4 – 2 + 4 = 8

Therefore,

Moves in part (ii) shows those steps who moves up

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