Understanding the nth Term
The nth term of the series is given by T(n) = 3n(n-1). This can be rewritten as:
- T(n) = 3n^2 - 3n
This represents a quadratic sequence, and we are interested in finding the sum of the first n terms, S(n).
Finding the Sum of the Series
To find the sum S(n), we need to calculate:
- S(n) = T(1) + T(2) + ... + T(n)
Substituting the nth term, we have:
- S(n) = Σ from k=1 to n of 3k(k-1)
This can be simplified as follows:
Breaking Down the Sum
The expression can further be expanded:
- S(n) = 3 * Σ from k=1 to n of (k^2 - k)
Now, we can separate the summations:
- S(n) = 3 * (Σ k^2 - Σ k)
Using known formulas for summation:
- Σ k^2 = n(n + 1)(2n + 1)/6
- Σ k = n(n + 1)/2
Final Expression for S(n)
Combining these results yields:
- S(n) = 3 * [n(n + 1)(2n + 1)/6 - n(n + 1)/2]
After simplifying, we arrive at:
- S(n) = n(n + 1)(n - 1)
This is the sum of the series for the first n terms where T(n) = 3n(n-1).
Conclusion
In conclusion, the sum of the series given the nth term 3n(n-1) is S(n) = n(n + 1)(n - 1). This formula can be used to find the total sum for any specified number of terms in the series.