Answer:
9
Step-by-step explanation:
The given set is that of a geometric sequence. We can identify this as each of the numbers in the set is equal to the product of the previous term by 2.
The nth term of any geometric sequence can be found by knowing:
In our case, we see that the first term is 7. We have also determined previously that the quotient is 2.
Therefore, the nth term of the sequence can be modeled using the formula
[tex]a_n = a_1 \times q^{n - 1}[/tex]
Where a1 is the first term in the sequence and q is the quotient. We'll substitute a1 = 7 and q = 2.
[tex]a_n = 7 \times 2^{n - 1}[/tex]
To find the cardinal number, we should find the natural number n for which [tex]a_n = 1792[/tex], as 1792 is the last term in the sequence. We'll substitute 1792 for [tex]a_n[/tex] and solve for n.
[tex]1792 = 7 \times 2^{n - 1} \text{ //}\div7\\256 = 2^{n - 1} \text{ //}\log_2()\\\log_2(256) = n - 1\\8 = n - 1 \text{ //}+1\\9 = n[/tex]
The cardinal number of the given set is 9.
