2.2 Consider the anithmetic sequence: $$ 5 ; 3 ; 1 ;-1 ;-3 ; \ldots ;-375 $$ 2.2.1 Write down $$ T_{6} $$. 2.2.2 Determine the $$ \mathrm{n}^{ ext {tb }} $$ term of the sequence. 2.2.3 Determine the number of terms in this sequence. 2.2.4 $$ \begin{array}{l} ext { There are } 64 ext { terms in this arithmetic sequence that are exactly } \ ext { divisible by } 3 ext {. Calculate their sum. }\end{array} $$ 2.2.5 Evaluate, without using a calculator: $$ \sum_{y=43^{\circ}}^{47^{\circ}} \sin ^{2} y $$

Question

2.2 Consider the anithmetic sequence: $$ 5 ; 3 ; 1 ;-1 ;-3 ; \ldots ;-375 $$ 2.2.1 Write down $$ T_{6} $$. 2.2.2 Determine the $$ \mathrm{n}^{	ext {tb }} $$ term of the sequence. 2.2.3 Determine the number of terms in this sequence. 2.2.4 $$ \begin{array}{l}	ext { There are } 64 	ext { terms in this arithmetic sequence that are exactly } \ 	ext { divisible by } 3 	ext {. Calculate their sum. }\end{array} $$ 2.2.5 Evaluate, without using a calculator: $$ \sum_{y=43^{\circ}}^{47^{\circ}} \sin ^{2} y $$

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### 2.2.1 Write down $$ T_{6} $$.

In an arithmetic sequence, the $$ n $$-th term can be calculated using the formula:
$$
T_n = a + (n-1)d
$$
where:
- $$ a $$ is the first term,
- $$ d $$ is the common difference,
- $$ n $$ is the term number.

From the sequence $$ 5, 3, 1, -1, -3, \ldots $$:
- The first term $$ a = 5 $$.
- The common difference $$ d = 3 - 5 = -2 $$.

Now, we can find $$ T_6 $$:
$$
T_6 = 5 + (6-1)(-2) = 5 + 5(-2) = 5 - 10 = -5
$$

### 2.2.2 Determine the $$ n^{\text{th}} $$ term of the sequence.

Using the same formula for the $$ n $$-th term:
$$
T_n = 5 + (n-1)(-2) = 5 - 2(n-1) = 5 - 2n + 2 = 7 - 2n
$$

### 2.2.3 Determine the number of terms in this sequence.

To find the number of terms, we need to determine when the last term $$ T_n = -375 $$:
$$
7 - 2n = -375
$$
Solving for $$ n $$:
$$
-2n = -375 - 7 \
-2n = -382 \
n = \frac{382}{2} = 191
$$

### 2.2.4 Calculate the sum of the 64 terms that are exactly divisible by 3.

First, we need to find the terms of the sequence that are divisible by 3. The general term is:
$$
T_n = 7 - 2n
$$
We want $$ T_n \equiv 0 \mod 3 $$:
$$
7 - 2n \equiv 0 \mod 3 \
2n \equiv 7 \mod 3 \
2n \equiv 1 \mod 3
$$
Multiplying both sides by the modular inverse of 2 modulo 3 (which is 2):
$$
n \equiv 2 \mod 3
$$
This means $$ n = 3k + 2 $$ for integers $$ k $$.

Now, we need to find the values of $$ n $$ that satisfy this condition and are within the range $$ 1 \leq n \leq 191 $$:
- The smallest $$ n $$ is 2 (when $$ k = 0 $$).
- The largest $$ n $$ is 191, which gives $$ k = 63 $$ (since $$ 3 \times 63 + 2 = 191 $$).

Thus, the terms divisible by 3 are $$ n = 2, 5, 8, \ldots, 191 $$. This is an arithmetic sequence with:
- First term $$ a = 2 $$,
- Common difference $$ d = 3 $$.

The number of terms $$ N $$ can be calculated as:
$$
N = \frac{191 - 2}{3} + 1 = \frac{189}{3} + 1 = 63 + 1 = 64
$$

Now, we can find the sum of these terms:
The sum $$ S $$ of an arithmetic sequence is given by:
$$
S = \frac{N}{2} (T_1 + T_N)
$$
where $$ T_1 = T_2 $$ and $$ T_N = T_{191} $$.

Calculating $$ T_2 $$ and $$ T_{191} $$:
$$
T_2 = 7 - 2(2) = 3 \
T_{191} = 7 - 2(191) = -375
$$

Now substituting into the sum formula:
$$
S = \frac{64}{2} (3 + (-375)) = 32 \times (-372) = -11904
$$

### 2.2.5 Evaluate $$ \sum_{y=43^{\circ}}^{47^{\circ}} \sin^2 y $$.

We can calculate this sum as follows:
$$
\sum_{y=43^{\circ}}^{47^{\circ}} \sin^2 y = \sin^2(43^{\circ}) + \sin^2(44^{\circ}) + \sin^2(45^{\circ}) + \sin^2(46^{\circ}) + \sin^2(47^{\circ})
$$

Using the known values:
- $$ \sin(45^{\circ}) = \frac{\sqrt{2}}{2} $$ so $$ \sin^2(45^{\circ}) = \frac{1}{2} $$.

For $$ \sin^2(43^{\circ}) $$ and $$ \sin^2(47^{\circ}) $$, we can use the identity $$ \sin^2(x) = \frac{1 - \cos(2x)}{2} $$ to find the values.

Calculating each term:
- $$ \sin^2(43^{\circ}) $$
- $$ \sin^2(44^{\circ}) $$
- $$ \sin^2(45^{\circ}) = \frac{1}{2} $$
- $$ \sin^2(46^{\circ}) $$
- $$ \sin^2(47^{\circ}) $$

Using symmetry, $$ \sin^2(43^{\circ}) = \sin^2(47^{\circ}) $$ and $$ \sin^2(44^{\circ}) = \sin^2(46^{\circ}) $$.

Thus, we can simplify the calculation:
$$
\sum_{y=43^{\circ}}^{47^{\circ}} \sin^2 y = 2\sin^2(43^{\circ}) + 2\sin^2(44^{\circ}) + \frac{1}{2}
$$

Calculating the exact values of $$ \sin^2(43^{\circ}) $$ and $$ \sin^2(44^{\circ}) $$ would typically require a calculator, but we can estimate or use known values.

Finally, the answer to each part is:
- **2.2.1:** $$ T_6 = -5 $$
- **2.2.2:** $$ T_n = 7 - 2n $$
- **2.2.3:** Number of terms = 191
- **2.2.4:** Sum of terms divisible by 3 = -11904
- **2.2.5:** $$ \sum_{y=43^{\circ}}^{47^{\circ}} \sin^
- **2.2.1:** The 6th term $$ T_6 $$ is -5. - **2.2.2:** The $$ n $$-th term of the sequence is $$ 7 - 2n $$. - **2.2.3:** There are 191 terms in the sequence. - **2.2.4:** The sum of the 64 terms divisible by 3 is -11,904. - **2.2.5:** The sum $$ \sum_{y=43^{\circ}}^{47^{\circ}} \sin^2 y $$ is approximately 5.

2.2 Consider the anithmetic sequence:

2.2.1 Write down .
2.2.2 Determine the term of the sequence.
2.2.3 Determine the number of terms in this sequence.
2.2.4
2.2.5 Evaluate, without using a calculator:

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2.2 Consider the anithmetic sequence: 2.2.1 Write down . 2.2.2 Determine the term of the sequence. 2.2.3 Determine the number of terms in this sequence. 2.2.4 2.2.5 Evaluate, without using a calculator: