Find the sum to n terms of the series whose nth term is
`4n^(3)+6n^(2)+2n`

Answer

To find the sum to n terms of the series whose nth term is given by \( T_n = 4n^3 + 6n^2 + 2n \), we will follow these steps:

### Step 1: Identify the nth term
The nth term of the series is given as:
\[ T_n = 4n^3 + 6n^2 + 2n \]

### Step 2: Calculate the first few terms
We will calculate the first term \( T_1 \) and the second term \( T_2 \).

- For \( n = 1 \):
\[ T_1 = 4(1)^3 + 6(1)^2 + 2(1) = 4 + 6 + 2 = 12 \]

- For \( n = 2 \):
\[ T_2 = 4(2)^3 + 6(2)^2 + 2(2) = 4(8) + 6(4) + 4 = 32 + 24 + 4 = 60 \]

### Step 3: Find the difference between consecutive terms
Now, we will find the difference \( d \) between the second term and the first term:
\[ d = T_2 - T_1 = 60 - 12 = 48 \]

### Step 4: Use the formula for the sum of n terms
The sum of the first n terms \( S_n \) of a series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

Here, \( a = T_1 = 12 \) and \( d = 48 \).

### Step 5: Substitute the values into the formula
Substituting the values into the sum formula:
\[ S_n = \frac{n}{2} \times (2 \times 12 + (n-1) \times 48) \]
\[ S_n = \frac{n}{2} \times (24 + (n-1) \times 48) \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ S_n = \frac{n}{2} \times (24 + 48n - 48) \]
\[ S_n = \frac{n}{2} \times (48n - 24) \]
\[ S_n = n \times (24n - 12) \]

### Final Result
Thus, the sum to n terms of the series is:
\[ S_n = 24n^2 - 12n \]

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