20
(Sigma Sign) 2n + 7
k= 1
∑2n+7
=2∑n + ∑ 7
=2*(20+1)(20/2) + 20*7
=420+140
=560
Thank you, MathMate.
Wait, do you know how it could be plugged into the Sn= a1 (1 - r^n/ 1 - r) formula?
Sn= a1 (1 - r^n/ 1 - r)
applies when the series is a geometric progression, such as
20
∑ a1*r^i
i=1
The expression you provided is in sigma notation. The sigma sign (∑) indicates a summation, and the expression inside the sigma is the term to be summed. In this case, the expression inside the sigma is 2n + 7.
The k=1 at the bottom of the sigma notation represents the starting value of k, which is 1 in this case. It indicates that we will start evaluating the sum from k = 1.
The 20 above the sigma sign is the upper limit of the sum, meaning we will evaluate the summation up to that value.
To find the value of this summation, you can manually evaluate it by substituting values of k starting from 1 and adding up the corresponding values of 2n + 7. Let's walk through the steps:
For k = 1:
2n + 7 = 2(1) + 7 = 2 + 7 = 9
For k = 2:
2n + 7 = 2(2) + 7 = 4 + 7 = 11
For k = 3:
2n + 7 = 2(3) + 7 = 6 + 7 = 13
Continue this process until you reach k = 20. Sum up all the values you obtain for each k, and that will be the value of the summation.
Alternatively, you can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
In this case, the first term is 2(1) + 7 = 9 and the last term is 2(20) + 7 = 47. The number of terms (n) is 20.
Sum = (20/2) * (9 + 47) = 10 * 56 = 560
Therefore, the value of the summation ∑(2n + 7) when k ranges from 1 to 20 is 560.