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An insect is on the 0 point of a number line, hopping towards 1. She covers half the distance from her current location to 1 with each hop. So, she will be at 1/2 after one hop, 3/4 after two hops, and so on.

(a) Make a table showing the insect’s location for the first 10 hops.
(b) Where will the insect be after n hops?
(c) Will the insect ever get to 1? Explain.

Solution:

Given, an insect is on the 0 point of a number line, hopping towards 1.

She covers half the distance from her current location to 1 with each hop.

So, she will be at 1/2 after one hop, 3/4 after two hops, and so on.

(a) We have to make a table showing the insect’s location for the first 10 hops.

Number of hops	Distance covered	Distance left	Distance covered
1	1/2	1/2	1- 1/2
2	1/2 (1/2) + 1/2	1/4	1 - 1/4
3	1/2 (1/4) + 3/4	1/8	1 - 1/8
4	1/2 (1/8) + 7/8	1/16	1 - 1/16
5	1/2 (1/16) + 15/16	1/32	1 - 1/32
6	1/2 (1/32) + 31/32	1/64	1 - 1/64
7	1/2 (1/64) + 63/64	1/128	1 - 1/128
8	1/2 (1/128) + 127/128	1/256	1 - 1/256
9	1/2 (1/256) + 255/256	1/512	1 - 1/512
10	1/2 (1/512) + 511/512	1/1024	1 - 1/1024

(b) Distance covered in first hop = 1 - 1/2

= 1 - (1/2)¹

Distance covered in second hop = 1 - (1/4)

= 1 - (1/2)²

Distance covered in third hop = 1 - (1/8)

= 1 - (1/2)³

We observe that the insect follows the pattern 1 - (1/2)ⁿ

(c) To get 1, (1/2)ⁿ has to be zero.

This is not possible because n is finite.

✦ Try This: Express each of the following in standard form: Express 8 tons in g.

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 12

NCERT Exemplar Class 8 Maths Chapter 8 Problem 127

An insect is on the 0 point of a number line, hopping towards 1. She covers half the distance from her current location to 1 with each hop. So, she will be at 1/2 after one hop, 3/4 after two hops, and so on. (a) Make a table showing the insect’s location for the first 10 hops, (b) Where will the insect be after n hops, (c) Will the insect ever get to 1? Explain

Summary:

(a) A table showing the insect’s location for the first 10 hops is mentioned above, (b) The insect after n hops will be there at 1 - (1/2)ⁿ, (c) The insect will never get to 1.

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