Question

Set builder form
C={2,4,8,16}

Answer 1

Consider the elements of the set C.

2, 4, 8, 16

Here we can find out that each term are multiplied up by 2.

2 × 2 = 4   ;   4 × 2 = 8   ;   8 × 2 = 16

Since each element is multiplied up by 2, we can say that the 4 terms are in a GP, geometric progression.

Now, what's about geometric progression?!

Let me explain geometric progression compared to arithmetic progression.

Like arithmetic progression in which terms are added up by the common difference, in geometric progression the terms are multiplied by the common ratio.

Like in arithmetic progression where the terms can be written in algebraic form a + (n - 1)d, in geometric progression the terms are written in algebraic form a · r^(n-1),  where r represents the common ratio and a is first term same as in AP.

In the case of 2, 4, 8, 16,  the first term is 2 and the common ratio is 2.

Hence the algebraic form will be  2 · 2^(n - 1) = 2^(1 + n - 1) = 2^n.

Yes, the elements can be expressed by the form 2^n. We can write this directly without having explanation about GP.

2¹ = 2   ;   2² = 4   ;   2³ = 8   ;   2⁴ = 16

And now we have to discuss about the values of 'n' given here.

Here 'n' is a natural number, the least value of 'n' given here is 1 and the highest value given here is 4.

Here the set C in set builder form will be,

$\Large\text{ C=\{x:x=2^n,\ n\in\mathbb{N},\ 1\leqslant n\leqslant 4\} }$