Certainly! Let's address each sequence step by step.
### Part (a)
The first sequence provided is:
[tex]\[
\frac{1}{3}, \frac{5}{6}, \frac{9}{9}, \frac{13}{12}, \frac{17}{15}, \ldots
\][/tex]
Step 1: Identify the general term
To find the general term \(a_n\), let's observe the pattern in the numerators and denominators separately.
- Numerator pattern: The numerators are \(1, 5, 9, 13, 17, \ldots\)
This appears to be an arithmetic sequence with the first term 1 and a common difference of \(4\):
[tex]\[
a_n = 4n - 3
\][/tex]
- Denominator pattern: The denominators are \(3, 6, 9, 12, 15, \ldots\)
This is an arithmetic sequence with the first term 3 and a common difference of \(3\):
[tex]\[
d_n = 3n
\][/tex]
Combining these observations, the general term for the sequence is:
[tex]\[
a_n = \frac{4n - 3}{3n}
\][/tex]
Step 2: Calculate the 6th and 7th terms
Using the general term \(a_n\):
- For \(n = 6\):
[tex]\[
a_6 = \frac{4(6) - 3}{3(6)} = \frac{24 - 3}{18} = \frac{21}{18} = \frac{7}{6}
\][/tex]
- For \(n = 7\):
[tex]\[
a_7 = \frac{4(7) - 3}{3(7)} = \frac{28 - 3}{21} = \frac{25}{21}
\][/tex]
Thus, the next two terms in the sequence are \(\frac{7}{6}\) and \(\frac{25}{21}\).
### Part (b)
The second sequence provided is:
[tex]\[
2, 7, 12, 17, 23, \ldots
\][/tex]
Step 1: Identify the general term
This sequence is strictly increasing by the same difference each time, making it an arithmetic sequence.
- The first term \(b_1 = 2\)
- The common difference \(d = 7 - 2 = 5\)
The general formula for the \(n\)-th term of an arithmetic sequence is:
[tex]\[
b_n = a + (n-1)d
\][/tex]
Substituting \(a = 2\) and \(d = 5\), we get:
[tex]\[
b_n = 2 + (n-1)5 = 2 + 5n - 5 = 5n - 3
\][/tex]
Step 2: Calculate the 6th and 7th terms
Using the general term \(b_n\):
- For \(n = 6\):
[tex]\[
b_6 = 5(6) - 3 = 30 - 3 = 27
\][/tex]
- For \(n = 7\):
[tex]\[
b_7 = 5(7) - 3 = 35 - 3 = 32
\][/tex]
Thus, the next two terms in the sequence are \(27\) and \(32\).
### Summary
For the sequences provided:
(a) The general term is:
[tex]\[
a_n = \frac{4n - 3}{3n}
\][/tex]
The next two terms after \(\frac{17}{15}\) are:
[tex]\[
\frac{7}{6}, \frac{25}{21}
\][/tex]
(b) The general term is:
[tex]\[
b_n = 5n - 3
\][/tex]
The next two terms after \(23\) are:
[tex]\[
27, 32
\][/tex]
