Find the sum in terms of n for Σ_k=1^n (3k^2-5k+7)
Not sure how to start working with this one
This might help:
https://brilliant.org/wiki/sum-of-n-n2-or-n3/
and thank you for the help with the Emily problem.
To find the sum in terms of n for the given series, Σ_k=1^n (3k^2 - 5k + 7), we can break it down into three separate summations and then apply the summation formulas for each term individually.
1. First, let's consider the summation Σ_k=1^n 3k^2. This is a sum of squares and can be evaluated using the formula for the sum of the first n squares, which is given by:
Σ_k=1^n k^2 = (n * (n + 1) * (2n + 1)) / 6
So, the summation Σ_k=1^n 3k^2 can be simplified as:
3 * Σ_k=1^n k^2 = 3 * [(n * (n + 1) * (2n + 1)) / 6]
2. Next, let's consider the summation Σ_k=1^n (-5k). This is a sum of arithmetic sequence, and we can use the formula for the sum of the first n terms of an arithmetic sequence, which is given by:
Σ_k=1^n a_k = n * (a_1 + a_n) / 2
In this case, a_k = -5k, so the summation Σ_k=1^n (-5k) can be simplified as:
-5 * Σ_k=1^n k = -5 * [n * (1 + n) / 2]
3. Finally, let's consider the summation Σ_k=1^n 7. This is a sum of a constant term, which simply means we are adding the same value (7) n number of times. So, the summation Σ_k=1^n 7 can be simplified as:
7 * Σ_k=1^n 1 = 7 * n
Now, we can add up all these simplifications to find the final sum:
Sum = 3 * [(n * (n + 1) * (2n + 1)) / 6] - 5 * [n * (1 + n) / 2] + 7 * n
Simplifying further will give you the sum in terms of n for the given series.