answer in terma of $$ y $$ QUENTION 2 The firat three forms of an ariflimetio sequence are 4; 13 and 22 1 Write down the fourth term of this sequence. 2 Determine the general torm of the sequence, 3 Consider the torms of this sequence which are even, Calculate tho sum of tho first 25 terms which are even.

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To solve the problem step by step, let's first extract the known conditions from the arithmetic sequence given:

1. The first three terms of the arithmetic sequence are $$ a_1 = 4 $$, $$ a_2 = 13 $$, and $$ a_3 = 22 $$.

### Step 1: Find the Fourth Term

The common difference $$ d $$ of an arithmetic sequence can be calculated as follows:

$$
d = a_2 - a_1 = 13 - 4 = 9
$$

Now, we can find the fourth term $$ a_4 $$:

$$
a_4 = a_3 + d = 22 + 9 = 31
$$

### Step 2: Determine the General Term of the Sequence

The general term $$ a_n $$ of an arithmetic sequence can be expressed as:

$$
a_n = a_1 + (n - 1) \cdot d
$$

Substituting the values we have:

$$
a_n = 4 + (n - 1) \cdot 9
$$

Simplifying this gives:

$$
a_n = 4 + 9n - 9 = 9n - 5
$$

### Step 3: Calculate the Sum of the First 25 Even Terms

To find the even terms of the sequence, we need to determine when $$ a_n $$ is even. The general term is:

$$
a_n = 9n - 5
$$

Setting $$ a_n $$ to be even:

$$
9n - 5 \equiv 0 \mod{2}
$$

This simplifies to:

$$
9n \equiv 5 \mod{2} \implies n \equiv 1 \mod{2}
$$

This means $$ n $$ must be odd. The odd integers are $$ 1, 3, 5, \ldots $$. The first 25 odd integers can be expressed as:

$$
n_k = 2k - 1 \quad \text{for } k = 1, 2, \ldots, 25
$$

Now, substituting $$ n_k $$ into the general term to find the even terms:

$$
a_{n_k} = 9(2k - 1) - 5 = 18k - 9 - 5 = 18k - 14
$$

Now, we need to calculate the sum of the first 25 even terms:

$$
S = \sum_{k=1}^{25} (18k - 14)
$$

This can be separated into two sums:

$$
S = 18 \sum_{k=1}^{25} k - 14 \cdot 25
$$

Using the formula for the sum of the first $$ n $$ integers:

$$
\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}
$$

For $$ n = 25 $$:

$$
\sum_{k=1}^{25} k = \frac{25 \cdot 26}{2} = 325
$$

Now substituting back into the sum $$ S $$:

$$
S = 18 \cdot 325 - 14 \cdot 25
$$

Calculating each term:

$$
S = 5850 - 350 = 5500
$$

### Final Answers

1. The fourth term of the sequence is $$ 31 $$.
2. The general term of the sequence is $$ a_n = 9n - 5 $$.
3. The sum of the first 25 even terms is $$ 5500 $$.
1. The fourth term is 31. 2. The general term is $$ a_n = 9n - 5 $$. 3. The sum of the first 25 even terms is 5500.

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