52582412
The sum of the first \(n\) terms of a sequence \((a_n)\) is \(S_n=2n^2+3n\).
1) Find \(a_1\), \(a_2\), and \(a_3\).
2) Derive an explicit formula for \(a_n\).
3) Show that \((a_n)\) is an arithmetic sequence.
Hints
- Recover a term by subtracting consecutive partial sums.
- Compute \(a_1\) directly from \(S_1\).
- A sequence is arithmetic when consecutive differences are constant.
Solution
1. Since \(a_1=S_1\),
\(a_1=2\cdot1^2+3\cdot1=5\).
Also,
\(a_2=S_2-S_1=14-5=9\),
and
\(a_3=S_3-S_2=27-14=13\).
2. In general,
\(a_n=S_n-S_{n-1}\)
\(=(2n^2+3n)-[2(n-1)^2+3(n-1)]=4n+1\).
3. Consecutive terms differ by
\(a_n-a_{n-1}=(4n+1)-[4(n-1)+1]=4\).
Since the difference is constant, the sequence is arithmetic.
Answer
1) \(a_1=5\), \(a_2=9\), and \(a_3=13\).
2) \(a_n=4n+1\).
3) The common difference is \(4\), so the sequence is arithmetic.
