Answer :
To solve the problem of finding which term [tex]\( m \)[/tex] in the arithmetic sequence [tex]\( 1, 9, 7, \ldots \)[/tex] is equal to 193, we should first understand how an arithmetic sequence works.
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (denoted as [tex]\( d \)[/tex]).
Given the terms [tex]\( 1, 9, 7 \)[/tex], this doesn't immediately seem to form a typical arithmetic sequence because [tex]\( 9 - 1 = 8 \)[/tex] and [tex]\( 7 - 9 = -2 \)[/tex]. This hints at alternating or inconsistent differences which typically complicate an arithmetic sequence.
However, assuming there might be a consistent pattern or a specific sequence that should be used, let's use a common difference [tex]\( d = 8 \)[/tex] that could hypothetically fit the sequence based on initially given terms.
Here's how you find which term [tex]\( m \)[/tex] is equal to 193:
1. Identify the first term [tex]\( a_1 \)[/tex]:
- [tex]\( a_1 = 1 \)[/tex]
2. Hypothetically assume a constant common difference [tex]\( d \)[/tex]:
- Assume [tex]\( d = 8 \)[/tex] based on adjusted pattern assumptions.
3. Use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \times d
\][/tex]
4. Set the [tex]\( n \)[/tex]-th term equal to 193 and solve for [tex]\( n \)[/tex] (which corresponds to [tex]\( m \)[/tex]):
[tex]\[
193 = 1 + (n - 1) \times 8
\][/tex]
5. Simplify the equation:
[tex]\[
193 - 1 = (n - 1) \times 8
\][/tex]
[tex]\[
192 = (n - 1) \times 8
\][/tex]
6. Divide by 8 to solve for [tex]\( n - 1 \)[/tex]:
[tex]\[
n - 1 = \frac{192}{8} = 24
\][/tex]
7. Add 1 to find [tex]\( n \)[/tex]:
[tex]\[
n = 24 + 1 = 25
\][/tex]
Thus, the 25th term of the hypothetical arithmetic sequence would be 193.
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (denoted as [tex]\( d \)[/tex]).
Given the terms [tex]\( 1, 9, 7 \)[/tex], this doesn't immediately seem to form a typical arithmetic sequence because [tex]\( 9 - 1 = 8 \)[/tex] and [tex]\( 7 - 9 = -2 \)[/tex]. This hints at alternating or inconsistent differences which typically complicate an arithmetic sequence.
However, assuming there might be a consistent pattern or a specific sequence that should be used, let's use a common difference [tex]\( d = 8 \)[/tex] that could hypothetically fit the sequence based on initially given terms.
Here's how you find which term [tex]\( m \)[/tex] is equal to 193:
1. Identify the first term [tex]\( a_1 \)[/tex]:
- [tex]\( a_1 = 1 \)[/tex]
2. Hypothetically assume a constant common difference [tex]\( d \)[/tex]:
- Assume [tex]\( d = 8 \)[/tex] based on adjusted pattern assumptions.
3. Use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \times d
\][/tex]
4. Set the [tex]\( n \)[/tex]-th term equal to 193 and solve for [tex]\( n \)[/tex] (which corresponds to [tex]\( m \)[/tex]):
[tex]\[
193 = 1 + (n - 1) \times 8
\][/tex]
5. Simplify the equation:
[tex]\[
193 - 1 = (n - 1) \times 8
\][/tex]
[tex]\[
192 = (n - 1) \times 8
\][/tex]
6. Divide by 8 to solve for [tex]\( n - 1 \)[/tex]:
[tex]\[
n - 1 = \frac{192}{8} = 24
\][/tex]
7. Add 1 to find [tex]\( n \)[/tex]:
[tex]\[
n = 24 + 1 = 25
\][/tex]
Thus, the 25th term of the hypothetical arithmetic sequence would be 193.
