To find the sum of the first 20 terms of the arithmetic progression (AP) given that the 10th term (\(a_{10}\)) is -5 and the 15th term (\(a_{15}\)) is -7.5, we start by using the general formula for the \(n\)th term of an AP:
\[ a_n = a + (n - 1)d \]
where \(a\) is the first term and \(d\) is the common difference.
From the information provided:
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The 10th term: \[ a_{10} = a + 9d = -5 \]
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The 15th term: \[ a_{15} = a + 14d = -7.5 \]
Now we can set up the two equations based on these terms:
\[ (1) \quad a + 9d = -5 \] \[ (2) \quad a + 14d = -7.5 \]
Next, we can subtract equation (1) from equation (2) to eliminate \(a\):
\[ (a + 14d) - (a + 9d) = -7.5 - (-5) \]
This simplifies to:
\[ 5d = -2.5 \]
Now, solve for \(d\):
\[ d = \frac{-2.5}{5} = -0.5 \]
Now that we have \(d\), we can substitute \(d\) back into equation (1) to find \(a\):
\[ a + 9(-0.5) = -5 \] \[ a - 4.5 = -5 \] \[ a = -5 + 4.5 = -0.5 \]
Now we have both \(a\) (the first term) and \(d\) (the common difference):
- \(a = -0.5\)
- \(d = -0.5\)
Next, we can find the sum of the first 20 terms of the AP. The formula for the sum of the first \(n\) terms \(S_n\) of an arithmetic sequence is given by:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \]
Setting \(n = 20\):
\[ S_{20} = \frac{20}{2} \cdot (2(-0.5) + (20 - 1)(-0.5)) \] \[ = 10 \cdot (2(-0.5) + 19(-0.5)) \] \[ = 10 \cdot (-1 + (-9.5)) \] \[ = 10 \cdot (-10.5) \] \[ = -105 \]
Therefore, the sum of the first 20 terms of the arithmetic progression is
\(\boxed{-105}\).