If where the given expression extends till infinity, which of the following statements must be true ?I. Two values are possible for zII. 4 - z2 = 2III. z8 = 16a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'D'. Can you explain this answer?

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where the given expression extends till infinity, which of the following statements must be true ?

I. Two values are possible for z

II. 4 - z² = 2

III. z⁸ = 16

a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None of the above

Correct answer is option 'D'. Can you explain this answer?

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If where the given expression extends till infinity, which of the foll...

Given:

To find: Which of the given 3 statements must be true about z?

Approach:

Looking at the 3 statements, we realize that in order to validate them, we’ll need to know the value of z
We’re given an expression for z². Using that expression, we’ll find the value of z

Working Out:

Finding the value of z
- Notice that the series of within within is never-ending. So, we can write:

To make our calculations easy, let’s replace z² with another variable, say 'y'

So, we get:
Now, squaring both sides:
y² = 2 + y
y² – y – 2 = 0
(y-2)(y+1) = 0
So, y = 2 or y = -1
- Since y = z² and perfect squares are never negative, y cannot be -1
So, y = z² = 2
This means, either z√2 or z-√2

Checking the validity of the 3 Statements
- Statement I: Two values are possible for z
- - As we’ve found above, indeed only 2 values are possible for z
  - Therefore, this statement is true
Statement II: 4 - z² = 2
- As we calculated above, z² = 2
- So, 4 – z² = 4 – 2 = 2
- So, this statement too is true
Statement III: z⁸ = 16
- Since z² = 2, (z²)⁴ = 2⁴
- So, z⁸ = 16
- Therefore, this statement is true as well
Looking at the answer choices, we see that the correct answer is Option D

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