This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the patterns and the word problem. Train Car Sequences Sequence D (red cars): 538, 547, 556, ___, ___, ___ Pattern: Add 9 to the previous number (547 - 538 = 9, 556 - 547 = 9). Next numbers: 556 + 9 = 565 565 + 9 = 574 574 + 9 = 583 The completed sequence is: 538, 547, 556, 565, 574, 583. Sequence C (blue cars): 729, 629, 529, ___, ___, ___ Pattern: Subtract 100 from the previous number (629 - 729 = -100, 529 - 629 = -100). Next numbers: 529 - 100 = 429 429 - 100 = 329 329 - 100 = 229 The completed sequence is: 729, 629, 529, 429, 329, 229. Sequence D (green cars): 347, 354, 361, ___, ___, ___ Pattern: Add 7 to the previous number (354 - 347 = 7, 361 - 354 = 7). Next numbers: 361 + 7 = 368 368 + 7 = 375 375 + 7 = 382 The completed sequence is: 347, 354, 361, 368, 375, 382. Complete the pattern. a) 373, 374, 376, ___, ___, ___ Pattern: Add 1, then add 2, then add 3, and so on. (The amount added increases by 1 each time). Next numbers: 376 + 3 = 379 379 + 4 = 383 383 + 5 = 388 The completed sequence is: 373, 374, 376, 379, 383, 388. b) 650, 653, 659, ___, ___, ___ Pattern: Add 3, then add 6, then add 12, and so on. (The amount added doubles each time). Next numbers: 659 + 12 = 671 671 + 24 = 695 695 + 48 = 743 The completed sequence is: 650, 653, 659, 671, 695, 743. c) 298, 303, 313, ___, ___, ___ Pattern: Add 5, then add 10, then add 20, and so on. (The amount added doubles each time). Next numbers: 313 + 20 = 333 333 + 40 = 373 373 + 80 = 453 The completed sequence is: 298, 303, 313, 333, 373, 453. d) 642, 644, 648, ___, ___, ___ Pattern: Add 2, then add 4, then add 8, and so on. (The amount added doubles each time). Next numbers: 648 + 8 = 656 656 + 16 = 672 672 + 32 = 704 The completed sequence is: 642, 644, 648, 656, 672, 704. e) 589, 593, 601, ___, ___, ___ Pattern: Add 4, then add 8, then add 16, and so on. (The amount added doubles each time). Next numbers: 601 + 16 = 617 617 + 32 = 649 649 + 64 = 713 The completed sequence is: 589, 593, 601, 617, 649, 713. f) 461, 467, 479, ___, ___, ___ Pattern: Add 6, then add 12, then add 24, and so on. (The amount added doubles each time). Next numbers: 479 + 24 = 503 503 + 48 = 551 551 + 96 = 647 The completed sequence is: 461, 467, 479, 503, 551, 647. Collecting shells James collects shells. Every day he picks up twice as many shells as the previous day. On the first day he picks up 7 shells. On the second day he picks up 14. How many shells would he collect on the ninth day? This is a geometric sequence where the first term a_1 = 7 and the common ratio r = 2. The formula for the n-th term of a geometric sequence is a_n = a_1 · r^n-1. Step 1: Identify the given values. a_1 = 7 (shells on the first day) r = 2 (twice as many as the previous day) n = 9 (ninth day) Step 2: Substitute the values into the formula. a_9 = 7 · 2^9-1 a_9 = 7 · 2^8 Step 3: Calculate 2^8. 2^8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 Step 4: Calculate a_9. a_9 = 7 · 256 a_9 = 1792 James would collect 1792 shells on the ninth day. That's 2 down. 3 left today — send the next one.