find the sum of the following arithmetic series and write in summation notation
a. 4, 11, ... to 16 terms
b. 19, 13, ... to 10 terms
thank you so much!
#1.
15
∑ 4+7k
k=0
a. To find the sum of the arithmetic series 4, 11, ... to 16 terms, we need to apply the formula for the sum of an arithmetic series:
S = (n/2)(a + l),
where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
Given that the first term (a) is 4, and we need to find the last term (l) for 16 terms, we use the formula to find the common difference:
d = (l - a)/(n - 1),
d = (l - 4)/(16 - 1).
To find the last term, we substitute the values of a and d into the equation for the n-th term:
l = a + (n - 1)d,
l = 4 + (16 - 1)d.
Now we have the values of a and l, and we can substitute them into the formula for the sum of an arithmetic series:
S = (n/2)(a + l),
S = (16/2)(4 + (16 - 1)d).
In summation notation, the sum of the arithmetic series is written as:
∑(i=1 to 16) (4 + (i - 1)d).
b. Similarly, to find the sum of the arithmetic series 19, 13, ... to 10 terms, we apply the same formula:
S = (n/2)(a + l).
Given that the first term (a) is 19, and we need to find the last term (l) for 10 terms, we find the common difference:
d = (l - a)/(n - 1),
d = (l - 19)/(10 - 1).
To find the last term, we substitute the values of a and d into the equation for the n-th term:
l = a + (n - 1)d,
l = 19 + (10 - 1)d.
Now we can substitute the values of a and l into the formula for the sum of an arithmetic series:
S = (n/2)(a + l),
S = (10/2)(19 + (10 - 1)d).
In summation notation, the sum of the arithmetic series is written as:
∑(i=1 to 10) (19 + (i - 1)d).
Please note that to find the actual sum, you would need to calculate the values of d and substitute them into the summation formulas.
To find the sum of an arithmetic series, you can use the formula:
Sn = (n/2) * (2a + (n-1)d)
where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.
a. For the series 4, 11, ..., to 16 terms:
In this case, the first term (a) is 4 and the common difference (d) is 11 - 4 = 7. We want to find the sum of 16 terms (n = 16).
Using the formula, we can plug in the values:
Sn = (16/2) * (2 * 4 + (16 - 1) * 7)
Simplifying further:
Sn = 8 * (8 + 15 * 7)
Sn = 8 * (8 + 105)
Sn = 8 * 113
Sn = 904
So, the sum of the arithmetic series is 904.
In summation notation, the series can be written as:
∑(i = 1 to 16) (4 + (i - 1) * 7)
b. For the series 19, 13, ..., to 10 terms:
Here, the first term (a) is 19 and the common difference (d) is 13 - 19 = -6. The number of terms is 10 (n = 10).
Using the formula, we can substitute the values:
Sn = (10/2) * (2 * 19 + (10 - 1) * -6)
Simplifying further:
Sn = 5 * (38 + 9 * -6)
Sn = 5 * (38 - 54)
Sn = 5 * (-16)
Sn = -80
Therefore, the sum of the arithmetic series is -80.
In summation notation, the series can be written as:
∑(i = 1 to 10) (19 + (i - 1) * -6)
a. a = 4 and d = 7
b, a = 19 and d = -6
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html