If for a sequence `{a_(n)},S_(n)=2n^(2)+9n`, where `S_(n)` is the sum of n terms, the value of `a_(20)` is

Options

A. 65
B. 75
C. 87
D. 97

Answer

To find the value of \( a_{20} \) for the sequence defined by the sum of the first \( n \) terms \( S_n = 2n^2 + 9n \), we can follow these steps:

### Step 1: Understand the relationship between \( S_n \) and \( a_n \)
The \( n \)-th term of the sequence \( a_n \) can be calculated using the formula:
\[ a_n = S_n - S_{n-1} \]
where \( S_n \) is the sum of the first \( n \) terms and \( S_{n-1} \) is the sum of the first \( n-1 \) terms.

### Step 2: Calculate \( S_n \) and \( S_{n-1} \)
Given:
\[ S_n = 2n^2 + 9n \]
Now, we need to find \( S_{n-1} \):
\[ S_{n-1} = 2(n-1)^2 + 9(n-1) \]
Expanding \( S_{n-1} \):
\[ S_{n-1} = 2(n^2 - 2n + 1) + 9(n - 1) = 2n^2 - 4n + 2 + 9n - 9 = 2n^2 + 5n - 7 \]

### Step 3: Calculate \( a_n \)
Now, substituting \( S_n \) and \( S_{n-1} \) into the formula for \( a_n \):
\[ a_n = S_n - S_{n-1} = (2n^2 + 9n) - (2n^2 + 5n - 7) \]
Simplifying this:
\[ a_n = 2n^2 + 9n - 2n^2 - 5n + 7 = 4n + 7 \]

### Step 4: Find \( a_{20} \)
Now, we can find \( a_{20} \):
\[ a_{20} = 4(20) + 7 = 80 + 7 = 87 \]

### Final Answer
Thus, the value of \( a_{20} \) is:
\[ \boxed{87} \]