In an arithmetic progression (A.P.), the \( n \)-th term is given by the formula:
\[ a_n = a + (n - 1)d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
Given:
- The first term \( a = 7 \)
- The 10th term is twice the second term.
The 10th term ( \( a_{10} \) ) can be calculated as follows:
\[ a_{10} = a + 9d = 7 + 9d \]
The second term ( \( a_2 \) ) is:
\[ a_2 = a + d = 7 + d \]
According to the problem, we have:
\[ a_{10} = 2a_2 \]
Substituting the expressions we have:
\[ 7 + 9d = 2(7 + d) \]
Expanding the right side:
\[ 7 + 9d = 14 + 2d \]
Now, isolate \( d \):
\[ 9d - 2d = 14 - 7 \]
\[ 7d = 7 \]
\[ d = 1 \]
Now that we have \( d = 1 \), we can find the relevant terms:
- Calculate the 19th term \( a_{19} \):
\[ a_{19} = a + (19 - 1)d = 7 + 18 \cdot 1 = 7 + 18 = 25 \]
- Sum of the 28th terms will be the same as the 28th term because it seems you mentioned "terms" instead of "terms". Thus, we will calculate \( a_{28} \):
\[ a_{28} = a + (28 - 1)d = 7 + 27 \cdot 1 = 7 + 27 = 34 \]
If you are looking for "the sum of the first 28 terms", that would be:
\[ S_n = \frac{n}{2} \cdot (a + a_n) \]
For the first 28 terms:
\[ S_{28} = \frac{28}{2} \cdot (7 + 34) = 14 \cdot 41 = 574 \]
- Calculate the difference between the 9th and 6th terms:
\[ a_{9} = a + (9-1)d = 7 + 8 \cdot 1 = 7 + 8 = 15 \]
\[ a_{6} = a + (6-1)d = 7 + 5 \cdot 1 = 7 + 5 = 12 \]
Now, the difference:
\[ a_{9} - a_{6} = 15 - 12 = 3 \]
Summary:
- The 19th term \( a_{19} \) is 25.
- The 28th term \( a_{28} \) is 34 (Or the sum of the first 28 terms is 574).
- The difference between the 9th and 6th terms is 3.