Question

question
given the sequence: 2,6,12,20,30,...
find the equation in standard form.
answer attempt 1 out of 4
additional solution no solution
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Explanation:

Step1: Analyze the sequence

Given sequence: \(2, 6, 12, 20, 30, \dots\)
Let's find the pattern by looking at the differences or the relationship with the term number \(n\) (where \(n = 1, 2, 3, \dots\))

For \(n = 1\), the term is \(2 = 1\times2\)
For \(n = 2\), the term is \(6 = 2\times3\)
For \(n = 3\), the term is \(12 = 3\times4\)
For \(n = 4\), the term is \(20 = 4\times5\)
For \(n = 5\), the term is \(30 = 5\times6\)

Step2: Derive the formula

We can see that for the \(n\)-th term, the value is \(n(n + 1)\)
Simplify \(n(n + 1)\):
\[

$$\begin{align*} n(n + 1)&=n^2 + n \end{align*}$$

\]

Answer:

The equation for the sequence is \(y = n^2 + n\) (where \(y\) represents the \(n\)-th term of the sequence)