Find the general term for the sequence pattern $2, 6, 12, 20, 30, \dots$

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Solution

Finding the general term for the sequence pattern
By observing the given sequence $2, 6, 12, 20, 30, \dots$ we get an idea that the pattern is quadratic
We can write each term as
$\begin{array}{rcl} 1^{2} + 1 & = & 2 \\ 2^{2} + 2 & = & 6 \\ 3^{2} + 3 & = & 12 \\ 4^{2} + 4 & = & 20 and so on . \end{array}$
$∴$ The general rule for the given sequence is $n^{2} + n, where n = 1, 2, 3, 4, \dots$

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2^{2} + 3^{2} + 6^{2} = 7^{2} 3^{2} + 4^{2} + 12^{2} = 13^{2} 4^{2} + 5^{2} + 20^{2} = 21^{2}

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