If the sum of first n terms of an A.P. is `3n^2 + 2n` , find its `r^(th)` term.

Answer

To find the \( r^{th} \) term of the arithmetic progression (A.P.) whose sum of the first \( n \) terms is given by \( S_n = 3n^2 + 2n \), we will follow these steps:

### Step 1: Understand the formula for the sum of the first \( n \) terms
The sum of the first \( n \) terms of an A.P. can be expressed as:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Find the first term \( a \)
To find \( a \), we can calculate \( S_1 \):
\[ S_1 = 3(1^2) + 2(1) = 3 + 2 = 5 \]
Since \( S_1 \) is the sum of the first term, we have:
\[ a = S_1 = 5 \]

**Hint for Step 1:** Substitute \( n = 1 \) into the sum formula to find the first term directly.

### Step 3: Find the second term \( a_2 \)
Next, we calculate \( S_2 \):
\[ S_2 = 3(2^2) + 2(2) = 3(4) + 4 = 12 + 4 = 16 \]
Now, the second term \( a_2 \) can be found using:
\[ a_2 = S_2 - S_1 = 16 - 5 = 11 \]

**Hint for Step 2:** Use the formula \( a_2 = S_2 - S_1 \) to find the second term.

### Step 4: Find the common difference \( d \)
The common difference \( d \) is given by:
\[ d = a_2 - a = 11 - 5 = 6 \]

**Hint for Step 3:** Calculate the common difference by subtracting the first term from the second term.

### Step 5: Write the formula for the \( r^{th} \) term
The \( r^{th} \) term of an A.P. is given by:
\[ a_r = a + (r-1)d \]
Substituting the values of \( a \) and \( d \):
\[ a_r = 5 + (r-1) \cdot 6 \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ a_r = 5 + 6r - 6 = 6r - 1 \]

### Final Answer
Thus, the \( r^{th} \) term of the A.P. is:
\[ \boxed{6r - 1} \]

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