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.

To find the fourth term of the arithmetic sequence, first determine the common difference by subtracting the first term from the second: \( 13 - 4 = 9 \). The fourth term can then be calculated as \( 22 + 9 = 31 \). Next, the general term of an arithmetic sequence is given by the formula \( a_n = a_1 + (n-1)d \). Here, \( a_1 = 4 \) and \( d = 9 \). Therefore, the general term is \( a_n = 4 + (n-1) \cdot 9 = 9n - 5 \). Now, for the even terms, we need to find when \( 9n - 5 \) is even. This occurs when \( n \) is odd (as \( 9n \) will be odd when \( n \) is odd and subtracting 5 will yield an even number). The odd terms are \( n = 1, 3, 5, \ldots \). The first 25 odd integers are \( 1, 3, 5, \ldots, 49 \). The corresponding even terms are \( a_1, a_3, a_5, \ldots, a_{49} \), which form another arithmetic sequence with first term \( a_1 = 4 \) and common difference \( d = 18 \) (every alternate term increases by 18). The sum of the first 25 even terms can be calculated with the formula for the sum of an arithmetic series: \[ S_{25} = \frac{n}{2} \cdot (2a + (n-1)d) \] Substituting \( n = 25 \), \( a = 4 \), and \( d = 18 \): \[ S_{25} = \frac{25}{2} \cdot (2 \cdot 4 + (25-1) \cdot 18) \\ = \frac{25}{2} \cdot (8 + 24 \cdot 18) \\ = \frac{25}{2} \cdot (8 + 432) \\ = \frac{25}{2} \cdot 440 \\ = 25 \cdot 220 = 5500 \] Thus, the sum of the first 25 even terms is \( 5500 \).