2. Three numbers form an arithmetic sequence whose common difference is 3. If the first number is increased by 1, the second increased by 6, and the third increased by 19, the resulting three numbers form a geometric sequence. Determine the original three numbers. 30 points 3. If x = 1, y = 7, and z = 15, determine a number that when added to x, y, and z yields consecutive terms of a geometric sequence. What are the first three terms in the geometric sequence? 30 points 4. Create your own word problem that starts with a three-number geometric sequence and after applying several operations then the results create an arithmetic sequence. Be sure to solve your word problem for full credit. 10 pointsMod 8 Honors Assignment - Sequences and Series Connecting Arithmetic and Geometric Sequences Example: Three numbers form a geometric sequence whose common ratio is 3. If the first is decreased by 1, the second decreased by 3, and the third reduced to 2 less than half its value, the resulting three numbers form an arithmetic sequence. Determine the original three numbers. Step 1: Represent the 3 numbers that form a x, 3x, 9x geometric sequence: Step 2: Represent the 3 numbers that form an x - 1, 3x - 3, _x - 2 arithmetic sequence: 2 Step 3: Apply the difference property of an arithmetic sequence: (3x -3) - (x -1) = 2x - 2 -(3x -3) Step 4: Solve for x x = 6 The original numbers are x =6, 3x =18, 9x = 54 Show and Explain all work for each problem 1. Three numbers form a geometric sequence whose common ratio is 0.5. If the first is reduced to 10 more than one quarter its value, the second decreased by 10, and the third increased by 10 more than twice its value, the resulting three numbers form an arithmetic sequence. Determine the original three numbers. 30 points