The sequence 5, 8, 11, 14, 17, 20 is called an arithmetic sequence because it has a common difference between consecutive terms. In this case, the common difference \(d\) is given by subtracting any term from the following term.
For example:
- \(8 - 5 = 3\)
- \(11 - 8 = 3\)
- \(14 - 11 = 3\)
- \(17 - 14 = 3\)
- \(20 - 17 = 3\)
Since the difference between each term is constant (3), the sequence is arithmetic.
Now, let's find the fifth term of the sequence using subscript notation:
- The first term \(a_1 = 5\)
- The second term \(a_2 = 8\)
- The third term \(a_3 = 11\)
- The fourth term \(a_4 = 14\)
- The fifth term \(a_5 = 17\)
Thus, we can state: \[ a_5 = 17 \]
Next, we can write an explicit formula for this arithmetic sequence. The general formula for the \(n\)-th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d \]
Where:
- \(a_1 = 5\) (the first term)
- \(d = 3\) (the common difference)
Plugging in these values, we have: \[ a_n = 5 + (n - 1) \cdot 3 \]
Now, we will use this formula to find the 50th term of the sequence \(a_{50}\).
Using the formula: \[ a_{50} = 5 + (50 - 1) \cdot 3 \]
Calculating: \[ a_{50} = 5 + (49) \cdot 3 \] \[ a_{50} = 5 + 147 \] \[ a_{50} = 152 \]
Putting it all together:
- \( a_n = 5 + (n - 1) \cdot 3 \)
- \( a_{50} = 5 + (50 - 1) \cdot 3 \)
- \( a_{50} = 5 + 49 \cdot 3 \)
- \( a_{50} = 152 \)
Therefore, the filled blanks are:
- Explicit formula: \( a_n = 5 + (n - 1) \cdot 3 \)
- 50th term calculation: \( a_{50} = 5 + (50 - 1) \cdot 3 \)
- Final result: \( a_{50} = 152 \)
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