To find the terms of the arithmetic progression (AP) given that the 10th term is \(-5\) and the 15th term is \(-7.5\), we can use the formula for the \(n\)th term of an AP, which is given by:
\[ a_n = a + (n - 1)d \]
Where:
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.
Given the information:
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For the 10th term: \[ a + 9d = -5 \quad \text{(1)} \]
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For the 15th term: \[ a + 14d = -7.5 \quad \text{(2)} \]
Now we can set up the equations based on the above two expressions.
Step 1: Solve the equations
From equation (1): \[ a + 9d = -5 \]
From equation (2): \[ a + 14d = -7.5 \]
Step 2: Subtract equation (1) from equation (2)
\[ (a + 14d) - (a + 9d) = -7.5 - (-5) \] \[ 14d - 9d = -7.5 + 5 \] \[ 5d = -2.5 \] \[ d = -\frac{2.5}{5} = -0.5 \]
Step 3: Substitute \(d\) back into one of the equations to find \(a\)
Substituting \(d = -0.5\) into equation (1): \[ a + 9(-0.5) = -5 \] \[ a - 4.5 = -5 \] \[ a = -5 + 4.5 = -0.5 \]
Conclusion
We find:
- \(a = -0.5\)
- \(d = -0.5\)
Resulting Terms
The general formula for the \(n\)th term becomes: \[ a_n = -0.5 + (n-1)(-0.5) \]
Calculating the first few terms:
- 1st term (\(a_1\)): \[ a_1 = -0.5 + 0 = -0.5 \]
- 2nd term (\(a_2\)): \[ a_2 = -0.5 - 0.5 = -1 \]
- 3rd term (\(a_3\)): \[ a_3 = -0.5 - 1 = -1.5 \]
Thus, the AP starts as: \[ -0.5, -1, -1.5, -2, \ldots \]
So the 10th and 15th terms are indeed \(-5\) and \(-7.5\), confirming our values for \(a\) and \(d\).