To find the arithmetic progression (AP) given the 7th term \( a_7 = 34 \) and the 20th term \( a_{20} = 113 \), 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 and \( d \) is the common difference.
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For the 7th term: \[ a_7 = a + 6d = 34 \quad \text{(Equation 1)} \]
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For the 20th term: \[ a_{20} = a + 19d = 113 \quad \text{(Equation 2)} \]
Now, we can solve these equations step-by-step.
Step 1: Subtract Equation 1 from Equation 2
\[ (a + 19d) - (a + 6d) = 113 - 34 \]
This simplifies to:
\[ 19d - 6d = 79 \] \[ 13d = 79 \] \[ d = \frac{79}{13} = 6.0769 \quad \text{(approximately)} \]
Step 2: Substitute \( d \) back into Equation 1 to find \( a \)
Substituting \( d \) into Equation 1:
\[ a + 6(6.0769) = 34 \]
Calculating \( 6 \times 6.0769 \):
\[ 6 \times 6.0769 \approx 36.4614 \] \[ a + 36.4614 = 34 \] \[ a = 34 - 36.4614 \] \[ a \approx -2.4614 \]
Final Form
The values of \( a \) and \( d \) are:
\[ a \approx -2.4614, \quad d \approx 6.0769 \]
General Term of the AP
The general term \( a_n \) can be expressed as:
\[ a_n = -2.4614 + (n-1) \cdot 6.0769 \]
Thus, the arithmetic progression is determined, and the first few terms can be calculated based on these values.