The sequence defined as s_n = 3n^3 + 2n + 1 appears in algebra and discrete problem sets, often checked on platforms like Chegg. Evaluating at n = 0 gives the initial term, which many learners use to verify their understanding of basic sequence notation.
By substituting n = 0 into the formula, students practice handling exponents, coefficients, and constant terms. This foundational exercise supports deeper work in series, limits, and function behavior for integer inputs.
| n | Term Formula | Expanded Calculation | Value s_n |
|---|---|---|---|
| 0 | 3n^3 + 2n + 1 | 3(0)^3 + 2(0) + 1 | 1 |
| 1 | 3n^3 + 2n + 1 | 3(1)^3 + 2(1) + 1 | 6 |
| 2 | 3n^3 + 2n + 1 | 3(8) + 4 + 1 | 29 |
| 3 | 3n^3 + 2n + 1 | 3(27) + 6 + 1 | 88 |
Evaluate s0 for the Given Cubic Sequence
To find s0, substitute n = 0 into the rule s_n = 3n^3 + 2n + 1. The cubic term 3(0)^3 is 0, the linear term 2(0) is 0, and the constant term remains 1. This straightforward substitution shows that the sequence starts at 1, which is often verified on homework platforms.
Interpret Sequence Behavior Near Zero
Examining values close to n = 0 helps learners see how each term contributes. At n = 0 only the constant survives, while at n = 1 and higher, the cubic growth quickly dominates. Understanding this shift supports graphing, limit concepts, and error checking in symbolic algebra tools.
Common Algebraic Steps and Pitfalls
When solving problems like this, students should carefully write each operation: cube n first, multiply by 3, compute 2n, then add 1. Errors often occur when signs or exponents are mishandled, so showing each step clearly reduces mistakes. Platforms like Chegg highlight these intermediate results to confirm correctness.
Connection to Series and Summation
Once s0 is known, learners can explore partial sums of the sequence to study series behavior. The initial term anchors the cumulative total, and comparing early partial sums reveals how fast the sequence grows. This foundation is useful in calculus, numerical methods, and algorithm analysis.
Key Takeaways for Working with Defined Sequences
- Always substitute the index value carefully, observing order of operations.
- Identify constant, linear, and higher-degree terms to see their individual contributions.
- Check simple cases like n = 0 and n = 1 to build intuition.
- Use these results to predict behavior and verify more complex calculations.
FAQ
Reader questions
How do I find s0 for s_n = 3n^3 + 2n + 1?
Substitute n = 0 into the formula: 3(0)^3 + 2(0) + 1 = 1, so s0 = 1.
What does s0 represent in a sequence?
It is the first term of the sequence, corresponding to the case when the index n equals zero.
Why is substitution a reliable method here?
The formula is explicitly defined for all integers n, so direct evaluation yields the exact term without approximation.
Can I verify this result using Chegg or similar tools?
Yes, entering the sequence definition and requesting s0 will typically show the step-by-step substitution and confirm the value 1.