To calculate \( S_{22} \), which is the sum of the first 22 terms of the arithmetic sequence, we can use the formula:
\[ S_n = \frac{n}{2} \times (a + a_n) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( a_n \) is the \( n \)-th term.
We are given \( a_{12} = 2.4 \) and the common difference \( d = 3.4 \). First, we need to find the first term \( a \) and the \( 22 \)-nd term \( a_{22} \).
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Find \( a_{12} \): The \( n \)-th term of the arithmetic sequence is given by: \[ a_n = a + (n-1) \cdot d \] For \( n = 12 \): \[ a_{12} = a + 11d \] Given \( a_{12} = 2.4 \) and \( d = 3.4 \): \[ 2.4 = a + 11(3.4) \] \[ 2.4 = a + 37.4 \] Therefore: \[ a = 2.4 - 37.4 = -35 \]
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Find \( a_{22} \): Now we can calculate the \( 22 \)-nd term: \[ a_{22} = a + 21d \] \[ a_{22} = -35 + 21(3.4) \] \[ a_{22} = -35 + 71.4 = 36.4 \]
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Calculate \( S_{22} \): Now we can plug \( n = 22 \), \( a = -35 \), and \( a_{22} = 36.4 \) into the sum formula: \[ S_{22} = \frac{22}{2} \times (-35 + 36.4) \] \[ S_{22} = 11 \times (1.4) \] \[ S_{22} = 15.4 \]
Thus, the calculated value of \( S_{22} \) is \( \boxed{15.4} \).