Question
Which of the following best describes this sequence? \( \{1,3,6,10,15,21, \ldots\} \) Arithmetic Geometric Both Neither
Ask by Floyd Simpson. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The sequence is neither arithmetic nor geometric.
Solution
To analyze the sequence {1, 3, 6, 10, 15, 21, …}, we first look at the differences between successive terms:
• 3 - 1 = 2
• 6 - 3 = 3
• 10 - 6 = 4
• 15 - 10 = 5
• 21 - 15 = 6
Since the differences are not constant, the sequence is not arithmetic.
Next, for a geometric sequence, the ratio between successive terms should be constant. Checking the ratios:
• 3 / 1 = 3
• 6 / 3 = 2
• 10 / 6 ≈ 1.67
• 15 / 10 = 1.5
• 21 / 15 = 1.4
Since these ratios are not equal, the sequence is not geometric either.
Therefore, the correct answer is: Neither.
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Simplify this solution Bonus Knowledge
This sequence is known as the triangular numbers, where each term represents a triangle with a given number of dots. The pattern is formed by adding consecutive natural numbers: \(1\), then \(1 + 2 = 3\), then \(3 + 3 = 6\), and so on. The \(n\)-th term of this sequence can be found using the formula \(\frac{n(n+1)}{2}\). In the world of math, sequences can inspire treasure hunts! Imagine a treasure map where each clue is hidden under a number of rocks corresponding to triangular numbers. As you tackle each clue, you gain insight into mathematical structures, and soon, you’ll see how these simple patterns open doors to understanding more complex concepts in algebra and beyond!
