Identify the sequence of even numbers from the given arithmetic progression.

Here are the solutions to the problems: 3. Calculate the sum of terms of the first 12 even numbers of an Arithmetic Progression: 1, 2, 3, 4,.... Step 1: Identify the sequence of even numbers from the given arithmetic progression. The given arithmetic progression is 1, 2, 3, 4, . The even numbers in this sequence are 2, 4, 6, 8, . This is an arithmetic progression with: First term, a = 2 Common difference, d = 4 - 2 = 2 Number of terms, n = 12 Step 2: Use the formula for the sum of an arithmetic progression. The sum of the first n terms of an arithmetic progression is given by: S_n = (n)/(2)(2a + (n-1)d) Substitute the values: a=2, d=2, n=12. S_12 = (12)/(2)(2(2) + (12-1)2) S_12 = 6(4 + (11)2) S_12 = 6(4 + 22) S_12 = 6(26) S_12 = 156 The sum of the first 12 even numbers is 156. 4. The range of a function g(x) = 2x - 1 is \-3, -1, 3, 0\. Draw an arrow showing the mapping with its domain. Step 1: Determine the domain values corresponding to the given range values. The function is g(x) = 2x - 1. The range is the set of output values, g(x). We need to find the input values, x, for each value in the range. Let y = g(x). So, y = 2x - 1. To find x, rearrange the equation: y + 1 = 2x x = (y+1)/(2) Step 2: Calculate the domain values for each range value. • For y = -3: x = (-3+1)/(2) = (-2)/(2) = -1 • For y = -1: x = (-1+1)/(2) = (0)/(2) = 0 • For y = 3: x = (3+1)/(2) = (4)/(2) = 2 • For y = 0: x = (0+1)/(2) = (1)/(2) So, the domain of the function is \-1, 0, 2, (1)/(2)\. Step 3: Draw an arrow diagram showing the mapping. Domain: \-1, 0, 2, (1)/(2)\ Range: \-3, -1, 3, 0\ The mapping is as follows: rcl Domain & & Range \\ -1 & & -3 \\ 0 & & -1 \\ 2 & & 3 \\ (1)/(2) & & 0