Sure, let's work step-by-step through the problem.
### Solving for the first term and common difference:
We are given:
- The sum of the first 7 terms of an arithmetic progression (AP) is 49.
- The sum of the first 17 terms of the AP is 289.
- Their product is 840 (though this information isn't directly needed for the calculations).
Using the formula for the sum of the first [tex]\( n \)[/tex] terms of an AP:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
For the sum of the first 7 terms:
[tex]\[ S_7 = 49 \][/tex]
[tex]\[ \frac{7}{2} [2a + 6d] = 49 \][/tex]
[tex]\[ 7 [a + 3d] = 49 \][/tex]
[tex]\[ a + 3d = 7 \quad \text{(Equation 1)} \][/tex]
For the sum of the first 17 terms:
[tex]\[ S_17 = 289 \][/tex]
[tex]\[ \frac{17}{2} [2a + 16d] = 289 \][/tex]
[tex]\[ 17 [a + 8d] = 289 \][/tex]
[tex]\[ a + 8d = 17 \quad \text{(Equation 2)} \][/tex]
Now, solve these simultaneous equations:
From Equation (1):
[tex]\[ a + 3d = 7 \][/tex]
From Equation (2):
[tex]\[ a + 8d = 17 \][/tex]
Subtract Equation (1) from Equation (2):
[tex]\[ (a + 8d) - (a + 3d) = 17 - 7 \][/tex]
[tex]\[ 5d = 10 \][/tex]
[tex]\[ d = 2 \][/tex]
Substitute [tex]\( d = 2 \)[/tex] into Equation (1):
[tex]\[ a + 3(2) = 7 \][/tex]
[tex]\[ a + 6 = 7 \][/tex]
[tex]\[ a = 1 \][/tex]
So, the first term [tex]\( a \)[/tex] is 1 and the common difference [tex]\( d \)[/tex] is 2.
### Arithmetic Series:
Now that we have [tex]\( a \)[/tex] and [tex]\( d \)[/tex], the arithmetic series is:
[tex]\[ 1, 3, 5, 7, 9, 11, 13, \ldots \][/tex]
### Sum of the first 20 terms:
Using the formula for the sum of the first [tex]\( n \)[/tex] terms of an AP:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
For [tex]\( n = 20 \)[/tex]:
[tex]\[ S_{20} = \frac{20}{2} [2 \cdot 1 + (20-1) \cdot 2] \][/tex]
[tex]\[ S_{20} = 10 [2 + 38] \][/tex]
[tex]\[ S_{20} = 10 \cdot 40 \][/tex]
[tex]\[ S_{20} = 400 \][/tex]
Therefore:
1. The first term ([tex]\( a \)[/tex]) is 1.
2. The common difference ([tex]\( d \)[/tex]) is 2.
3. The arithmetic series is: [tex]\( 1, 3, 5, 7, 9, 11, 13, \ldots \)[/tex]
4. The sum of the first 20 terms is 400.
