Absolutely, let's work through this step-by-step.
Given the sequence: \(27, 40, 53\),
the next two terms are \(92\) and \(105\).
Firstly, let's identify the pattern in the sequence. We'll look for a common difference since this appears to be an arithmetic sequence.
1. Calculate the common difference:
[tex]\[
d = 40 - 27 = 13
\][/tex]
[tex]\[
d = 53 - 40 = 13
\][/tex]
So, the common difference \(d\) is indeed \(13\).
2. To confirm the consistency, even though the sequence breaks between the given terms, let’s continue with the known common difference:
- After \(53\), the next term \(= 53 + 13 = 66\)
- After \(66\), the next term \(= 66 + 13 = 79\)
- Continuing this pattern:
- After \(79\), the next term \(= 79 + 13 = 92\) (which confirms the next term provided)
3. Now, with the common difference verified, we can calculate the 20th term, even though it is directly given.
This is more for verifying the constant sequence we calculated is accurate:
[tex]\[
a_{20} = a_1 + (20-1)d = 27 + 19\times 13 = 27 + 247 = 274
\][/tex]
So, the 20th term is indeed \(274\).
4. Finally, calculate the 21st term:
[tex]\[
a_{21} = a_{20} + d = 274 + 13 = 287
\][/tex]
Hence, the twenty-first term of the sequence is [tex]\(\boxed{287}\)[/tex].
