The sum of first n terms of an A.P. is `5n^(2) + 3n`. If the nth term is 168, find the value of n. Also find the 20th term of the A.P.

Answer

To solve the problem step by step, we will follow the given information and use the formulas related to the Arithmetic Progression (A.P.).

### Step 1: Understand the given information
We are given that the sum of the first \( n \) terms of an A.P. is \( S_n = 5n^2 + 3n \) and the \( n \)th term \( a_n = 168 \).

### Step 2: Find the first term \( a_1 \) and second term \( a_2 \)
The sum of the first term \( S_1 \) is equal to the first term \( a_1 \):
\[ S_1 = 5(1)^2 + 3(1) = 5 + 3 = 8 \implies a_1 = 8 \]

The sum of the first two terms \( S_2 \) is:
\[ S_2 = 5(2)^2 + 3(2) = 5(4) + 6 = 20 + 6 = 26 \]
Thus, we can express \( S_2 \) as:
\[ S_2 = a_1 + a_2 \implies 26 = 8 + a_2 \implies a_2 = 26 - 8 = 18 \]

### Step 3: Identify the first term \( a \) and common difference \( d \)
Now we have:
- \( a_1 = 8 \)
- \( a_2 = 18 \)

The common difference \( d \) can be calculated as:
\[ d = a_2 - a_1 = 18 - 8 = 10 \]

### Step 4: Write the formula for the \( n \)th term
The formula for the \( n \)th term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ a_n = 8 + (n - 1) \cdot 10 \]
This simplifies to:
\[ a_n = 10n - 2 \]

### Step 5: Set the \( n \)th term equal to 168
From the problem, we know \( a_n = 168 \):
\[ 10n - 2 = 168 \]
Adding 2 to both sides:
\[ 10n = 170 \]
Dividing both sides by 10:
\[ n = 17 \]

### Step 6: Find the 20th term of the A.P.
Now, we need to find the 20th term \( a_{20} \):
\[ a_{20} = a + (20 - 1) \cdot d = 8 + 19 \cdot 10 \]
Calculating this gives:
\[ a_{20} = 8 + 190 = 198 \]

### Final Answers
- The value of \( n \) is \( 17 \).
- The 20th term of the A.P. is \( 198 \).