use summation notation to write each arithmetic series for the specifies number of terms. Then evaluate the sum. 10+7+4+...; n=1
1 term, so the sum is just the 1st term: 10
Actually, we need to write the arithmetic series using summation notation first. The given series is:
10 + 7 + 4 + ...
The common difference between consecutive terms is -3 (subtracting 3 each time).
To write it using summation notation, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms.
We know the first term a_1 = 10 and the common difference d = -3. We want to write the series for n = 1 term.
So, substituting these values in the formula, we get:
a_1 + (n-1)d = 10 + (1-1)(-3) = 10
Therefore, the series for n = 1 term is simply:
a_1 = 10
And the sum of this series is also 10.
To write an arithmetic series using summation notation, we need to determine the common difference (d) and the initial term (a₁).
Given the arithmetic series: 10 + 7 + 4 + ...
To find the common difference, we subtract each term from the previous term:
7 - 10 = -3
4 - 7 = -3
Hence, the common difference (d) is -3.
To find the initial term (a₁), we use the first term of the series:
a₁ = 10
Now, we can write the arithmetic series in summation notation:
Sum from n=1 to infinity of aₙ = a₁ + (n-1)d
Substituting the values:
Sum from n=1 to infinity of aₙ = 10 + (n-1)(-3)
Now, if we want to evaluate the sum for a specific number of terms, we substitute the value of n.
For n = 1:
Sum from n=1 to 1 of aₙ = 10 + (1-1)(-3)
Sum from n=1 to 1 of aₙ = 10 + 0 = 10
So, when n = 1, the sum of the series is 10.
To write the arithmetic series using summation notation, we need to determine the general term of the series and the upper limit of the summation.
The given arithmetic series starts with 10, and each subsequent term decreases by 3 (7 = 10 - 3, 4 = 7 - 3, and so on). Hence, the general term of the series can be expressed as:
a(n) = 10 + (n-1)(-3)
Where "n" represents the term number in the series.
Now, we can use the given value of n = 1 to evaluate the sum of the series. Plugging in n = 1 into the general term equation, we have:
a(1) = 10 + (1-1)(-3)
= 10 + 0
= 10
To evaluate the sum of the series, we can use the summation notation, which is represented by the symbol Σ (sigma). The upper limit of the summation is given by n = 1 in this case.
The summation notation for the arithmetic series is:
Σ(a(n)) = Σ(10 + (n-1)(-3)), where the summation is from n = 1 to n = 1.
Substituting n = 1 into the general term equation, we have:
Σ(a(n)) = Σ(10 + (1-1)(-3))
= Σ(10)
Since there is only one term in the series (n = 1), the sum of the series is simply the value of the single term, which is 10.