Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.
(i) 16, 19, 22, 25
(ii) 1, 5, 12, 22, 35, 51, 70, 92, 117, 145
Explanation
The student identifies the recursive rule from the context that each term is 3 more than the previous term to find the next four terms. Then, they apply the cumulative addition rule as hinted in the textbook to derive the first 10 terms of the new sequence.
Solution Steps
Step 1: Identify the pattern of the given sequence (each term is 3 more than the previous term).
Step 2: Calculate the next four terms by adding 3 successively: 13+3=16, 16+3=19, 19+3=22, 22+3=25.
Step 3: List the first 10 terms of the original sequence: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28.
Step 4: Calculate the cumulative sum to find the first 10 terms of the new sequence: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.