To determine whether the given set of numbers [tex]\(0, 1, 3, 6, 10, 15, 21, \ldots\)[/tex] forms a sequence, we need to analyze the pattern and formula that defines the sequence.
1. Identify the given sequence:
- The given numbers are [tex]\(0, 1, 3, 6, 10, 15, 21, \ldots\)[/tex].
2. Observe the numbers and check for a pattern:
- Let's look at the differences between consecutive terms:
[tex]\[
\begin{align*}
1 - 0 &= 1, \\
3 - 1 &= 2, \\
6 - 3 &= 3, \\
10 - 6 &= 4, \\
15 - 10 &= 5, \\
21 - 15 &= 6.
\end{align*}
\][/tex]
- The differences between consecutive terms are [tex]\(1, 2, 3, 4, 5, 6, \ldots\)[/tex].
3. Check if the pattern matches a known sequence:
- The differences [tex]\(1, 2, 3, 4, 5, 6, \ldots\)[/tex] indicate that each term increases by the next integer in the sequence.
4. Identify the sequence formula:
- These differences are characteristic of the triangular numbers, which can be expressed by the formula:
[tex]\[
T_n = \frac{n(n + 1)}{2}
\][/tex]
- Now, let's verify each term with this triangular number formula:
[tex]\[
\begin{align*}
T_0 &= \frac{0(0 + 1)}{2} = 0, \\
T_1 &= \frac{1(1 + 1)}{2} = 1, \\
T_2 &= \frac{2(2 + 1)}{2} = 3, \\
T_3 &= \frac{3(3 + 1)}{2} = 6, \\
T_4 &= \frac{4(4 + 1)}{2} = 10, \\
T_5 &= \frac{5(5 + 1)}{2} = 15, \\
T_6 &= \frac{6(6 + 1)}{2} = 21.
\end{align*}
\][/tex]
5. Conclusion:
- Each term in the provided set of numbers matches the corresponding triangular number given by [tex]\(T_n = \frac{n(n + 1)}{2}\)[/tex].
- Therefore, the given set of numbers [tex]\(0, 1, 3, 6, 10, 15, 21, \ldots\)[/tex] conforms to the formula of triangular numbers, indicating it is a sequence.
Hence, the given set of numbers is indeed a number sequence, specifically the sequence of triangular numbers.
