The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Determine the general term of the sequence.
Question 1 Report
The sum of the first n terms of a linear sequence is \(S_{n} = n^{2} + 2n\). Determine the general term of the sequence.
Answer Details
To find the general term of a linear sequence, we need to find the common difference between any two consecutive terms of the sequence.
We know that the sum of the first n terms of the sequence is given by:
$$S_{n} = n^{2} + 2n$$
If we subtract the sum of the first (n-1) terms from the sum of the first n terms, we get the nth term of the sequence:
$$\begin{aligned} a_{n} &= S_{n} - S_{n-1} \\ &= (n^{2} + 2n) - [(n-1)^{2} + 2(n-1)] \\ &= 2n - 1 \end{aligned}$$
Therefore, the general term of the sequence is given by:
$$a_{n} = 2n - 1$$
So, the correct option is \textbf{2n + 1}.
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