Sure, let's solve the given problem step by step.
Given:
- The sum of the first 14 terms of an arithmetic progression (AP) is 1050.
- The first term (a) of the AP is 2.
We need to find:
- The common difference (d) of the AP.
- The 20th term (a_20) of the AP.
### Step 1: Determine the Common Difference (d)
The sum of the first [tex]\( n \)[/tex] terms of an AP is given by the formula:
[tex]\[ S_n = \frac{n}{2} \left( 2a + (n - 1)d \right) \][/tex]
For the first 14 terms:
[tex]\[ S_{14} = 1050 \][/tex]
[tex]\[ a = 2 \][/tex]
[tex]\[ n = 14 \][/tex]
Substituting these values into the formula:
[tex]\[ 1050 = \frac{14}{2} \left( 2 \cdot 2 + (14 - 1)d \right) \][/tex]
Simplify:
[tex]\[ 1050 = 7 \left( 4 + 13d \right) \][/tex]
Solving for [tex]\( d \)[/tex]:
[tex]\[ 1050 = 28 + 91d \][/tex]
[tex]\[ 1050 - 28 = 91d \][/tex]
[tex]\[ 1022 = 91d \][/tex]
Now, divide both sides by 91:
[tex]\[ d = \frac{1022}{91} \][/tex]
The common difference [tex]\( d \)[/tex] is approximately:
[tex]\[ d \approx 11.2308 \][/tex]
### Step 2: Find the 20th Term (a_20)
The [tex]\( n \)[/tex]-th term of an AP is given by:
[tex]\[ a_n = a + (n - 1)d \][/tex]
To find the 20th term ([tex]\( a_{20} \)[/tex]):
[tex]\[ n = 20 \][/tex]
[tex]\[ a_{20} = a + (20 - 1)d \][/tex]
Substitute [tex]\( a = 2 \)[/tex] and [tex]\( d \approx 11.2308 \)[/tex] into the formula:
[tex]\[ a_{20} = 2 + (20 - 1) \cdot 11.2308 \][/tex]
[tex]\[ a_{20} = 2 + 19 \cdot 11.2308 \][/tex]
[tex]\[ a_{20} = 2 + 213.3846 \][/tex]
[tex]\[ a_{20} \approx 215.3846 \][/tex]
### Conclusion
The common difference [tex]\( d \)[/tex] of the arithmetic progression is approximately 11.2308, and the 20th term [tex]\( a_{20} \)[/tex] is approximately 215.3846.
