To solve this problem, we will find the 53rd term of the given geometric sequence and then compute the logarithm base 2 of this term. Here's the detailed step-by-step process:
1. Identify the first term and common ratio:
The sequence provided is [tex]\(8, 16, 32, 64, \ldots\)[/tex].
- The first term ([tex]\(a_1\)[/tex]) of the sequence is 8.
- To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term: [tex]\( r = \frac{16}{8} = 2 \)[/tex].
2. General formula for the n-th term of a geometric sequence:
The formula for the n-th term ([tex]\(a_n\)[/tex]) of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
3. Calculate the 53rd term:
Here, we need to find the 53rd term ([tex]\(a_{53}\)[/tex]):
[tex]\[
a_{53} = 8 \cdot 2^{(53-1)} = 8 \cdot 2^{52}
\][/tex]
4. Value of the 53rd term:
After computing, [tex]\(a_{53} = 36028797018963968\)[/tex].
5. Compute [tex]\(\log_2(x)\)[/tex]:
To find [tex]\(\log_2(x)\)[/tex], where [tex]\(x\)[/tex] is the 53rd term:
[tex]\[
\log_2(36028797018963968)
\][/tex]
Since [tex]\(36028797018963968\)[/tex] can be expressed as [tex]\(2^{55}\)[/tex]:
[tex]\[
\log_2(2^{55}) = 55
\][/tex]
Hence, the value of [tex]\(\log_2(x)\)[/tex] is 55.
Therefore, the 53rd term in the sequence is [tex]\(36028797018963968\)[/tex], and [tex]\(\log_2\)[/tex] of this term is [tex]\(55\)[/tex].
