Consider a sequence whose sum to n terms is given by the quadratic function Sn= 3(n^2) +5n. Then sum of the squares of the first 3 terms of the given series is

Options

A. 1100
B. 660
C. 799
D. 1000

Explanation

To solve the problem, we need to find the sum of the squares of the first three terms of the sequence whose sum to n terms is given by the quadratic function \( S_n = 3n^2 + 5n \).

### Step 1: Find the nth term \( T_n \)

The nth term \( T_n \) can be found using the formula:
\[ T_n = S_n - S_{n-1} \]
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 \( S_{n-1} \)

To find \( S_{n-1} \), we substitute \( n-1 \) into the sum formula:
\[ S_{n-1} = 3(n-1)^2 + 5(n-1) \]
Expanding this:
\[ S_{n-1} = 3(n^2 - 2n + 1) + 5(n - 1) = 3n^2 - 6n + 3 + 5n - 5 = 3n^2 - n - 2 \]

### Step 3: Substitute \( S_n \) and \( S_{n-1} \) into the formula for \( T_n \)

Now, substituting \( S_n \) and \( S_{n-1} \):
\[ T_n = (3n^2 + 5n) - (3n^2 - n - 2) \]
Simplifying this:
\[ T_n = 3n^2 + 5n - 3n^2 + n + 2 = 6n + 2 \]

### Step 4: Calculate the first three terms \( T_1, T_2, T_3 \)

Now we can find the first three terms:
- For \( n = 1 \):
\[ T_1 = 6(1) + 2 = 6 + 2 = 8 \]
- For \( n = 2 \):
\[ T_2 = 6(2) + 2 = 12 + 2 = 14 \]
- For \( n = 3 \):
\[ T_3 = 6(3) + 2 = 18 + 2 = 20 \]

### Step 5: Calculate the sum of the squares of the first three terms

Now we need to find \( T_1^2 + T_2^2 + T_3^2 \):
\[ T_1^2 = 8^2 = 64 \]
\[ T_2^2 = 14^2 = 196 \]
\[ T_3^2 = 20^2 = 400 \]
Now, summing these:
\[ T_1^2 + T_2^2 + T_3^2 = 64 + 196 + 400 = 660 \]

### Final Answer
The sum of the squares of the first three terms is \( 660 \).

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