Suppose `a_(1)=45, a_(2)=41 and a_(k)=2a_(k-1)-a_(k-2) AA k ge 3`, then, value of
`S=(1)/(12) [a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(11)^(2)-a_(12)^(2)]` is

Options

A. 89
B. 90
C. 91
D. 92

Answer

To solve the problem step by step, we will follow the instructions given in the question and use the recurrence relation to find the values of \( a_k \) for \( k = 3 \) to \( 12 \). Then we will compute the value of \( S \).

### Step 1: Calculate the first few terms of the sequence

Given:
- \( a_1 = 45 \)
- \( a_2 = 41 \)
- The recurrence relation: \( a_k = 2a_{k-1} - a_{k-2} \) for \( k \geq 3 \)

Let's calculate the next terms:

- **For \( k = 3 \)**:
\[ a_3 = 2a_2 - a_1 = 2 \times 41 - 45 = 82 - 45 = 37 \]

- **For \( k = 4 \)**:
\[ a_4 = 2a_3 - a_2 = 2 \times 37 - 41 = 74 - 41 = 33 \]

- **For \( k = 5 \)**:
\[ a_5 = 2a_4 - a_3 = 2 \times 33 - 37 = 66 - 37 = 29 \]

- **For \( k = 6 \)**:
\[ a_6 = 2a_5 - a_4 = 2 \times 29 - 33 = 58 - 33 = 25 \]

- **For \( k = 7 \)**:
\[ a_7 = 2a_6 - a_5 = 2 \times 25 - 29 = 50 - 29 = 21 \]

- **For \( k = 8 \)**:
\[ a_8 = 2a_7 - a_6 = 2 \times 21 - 25 = 42 - 25 = 17 \]

- **For \( k = 9 \)**:
\[ a_9 = 2a_8 - a_7 = 2 \times 17 - 21 = 34 - 21 = 13 \]

- **For \( k = 10 \)**:
\[ a_{10} = 2a_9 - a_8 = 2 \times 13 - 17 = 26 - 17 = 9 \]

- **For \( k = 11 \)**:
\[ a_{11} = 2a_{10} - a_9 = 2 \times 9 - 13 = 18 - 13 = 5 \]

- **For \( k = 12 \)**:
\[ a_{12} = 2a_{11} - a_{10} = 2 \times 5 - 9 = 10 - 9 = 1 \]

Now we have:
- \( a_1 = 45 \)
- \( a_2 = 41 \)
- \( a_3 = 37 \)
- \( a_4 = 33 \)
- \( a_5 = 29 \)
- \( a_6 = 25 \)
- \( a_7 = 21 \)
- \( a_8 = 17 \)
- \( a_9 = 13 \)
- \( a_{10} = 9 \)
- \( a_{11} = 5 \)
- \( a_{12} = 1 \)

### Step 2: Calculate \( S \)

The formula for \( S \) is given as:
\[ S = \frac{1}{12} \left( a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{11}^2 - a_{12}^2 \right) \]

We will compute each term:
- \( a_1^2 = 45^2 = 2025 \)
- \( a_2^2 = 41^2 = 1681 \)
- \( a_3^2 = 37^2 = 1369 \)
- \( a_4^2 = 33^2 = 1089 \)
- \( a_5^2 = 29^2 = 841 \)
- \( a_6^2 = 25^2 = 625 \)
- \( a_7^2 = 21^2 = 441 \)
- \( a_8^2 = 17^2 = 289 \)
- \( a_9^2 = 13^2 = 169 \)
- \( a_{10}^2 = 9^2 = 81 \)
- \( a_{11}^2 = 5^2 = 25 \)
- \( a_{12}^2 = 1^2 = 1 \)

Now, substituting these values into the formula for \( S \):
\[ S = \frac{1}{12} \left( 2025 - 1681 + 1369 - 1089 + 841 - 625 + 441 - 289 + 169 - 81 + 25 - 1 \right) \]

Calculating the expression inside the parentheses:
\[ = \frac{1}{12} \left( 2025 - 1681 + 1369 - 1089 + 841 - 625 + 441 - 289 + 169 - 81 + 25 - 1 \right) \]
\[ = \frac{1}{12} \left( 2025 - 1681 = 344 \right) \]
\[ = \frac{1}{12} \left( 344 + 1369 - 1089 = 280 \right) \]
\[ = \frac{1}{12} \left( 280 + 841 - 625 = 496 \right) \]
\[ = \frac{1}{12} \left( 496 + 441 - 289 = 648 \right) \]
\[ = \frac{1}{12} \left( 648 + 169 - 81 = 736 \right) \]
\[ = \frac{1}{12} \left( 736 + 25 - 1 = 760 \right) \]
\[ = \frac{760}{12} = 63.33 \]

Thus, the final value of \( S \) is:
\[ S = 63.33 \]