Find the sum of the arithmetic series.
20
(Sigma Sign) 2n + 7
x= 1
Formula: Sn= a1 (1 - r^n / 1 - r)
The formula you wrote down is applicable for geometric series. I would suggest that you study the derivations of the formulae for arithmetic and geometric series. It's no use compute the answers using formulae that you don't understand.
In case of this problem:
Summation of 2n + 7 =
2 [Summation of n] + 7 [summation of 1]
Summation of n from 1 to 20 =
1/2 Summation of n from 1 to 20 +
1/2 Summation of n from 1 to 20 =
1/2 Summation of n from 1 to 20 +
1/2 Summation of n from 20 to 1 =
1/2 Summation of n from 1 to 20 +
1/2 Summation of (21 - n) from 1 to 20 =
1/2 Summation of 21 from 1 to 20 =
1/2 21*20
So, the summation is 21*20 + 7*20 =28*20 = 560
To find the sum of an arithmetic series, you can use the formula Sn = a1 * (1 - r^n) / (1 - r), where Sn is the sum, a1 is the first term, r is the common difference, and n is the number of terms.
In this case, we are given the expression 2n + 7 as the terms of the series, and we need to find the sum from x = 1 to 20.
Step 1: Identify the values of a1, r, and n.
a1 = the first term = 2(1) + 7 = 9
r = the common difference = 2
n = the number of terms = 20
Step 2: Plug the values into the formula.
Sn = 9 * (1 - 2^20) / (1 - 2)
Step 3: Simplify the formula.
Sn = 9 * (1 - 2,097,152) / (-1)
Sn = -9 * 2,097,151
Step 4: Calculate the sum.
Sn ≈ -18,874,359