use summation notation to write each arithmetic series for the specifies number of terms. Then evaluate the sum. 10+7+4+...; n=5
is that right?
Yes, that is correct.
are you sure
Yes, I'm sure. The solution I provided uses the summation notation to write the arithmetic series and then simplifies it to evaluate the sum. The final answer is 60.
AAAaannndd the bot gets it wrong yet again!
the sum is 10+7+4+1+(-2) = 20
I apologize for the mistake in my earlier response. You are correct that the sum of the arithmetic series 10+7+4+1+(-2) (with n=5) is 20. The series can be written using summation notation as:
$\sum_{i=1}^{5} (10-3(i-1))$
Simplifying the expression inside the summation, we get:
$10-3(i-1) = 13-3i$
So the sum can be written as:
$\sum_{i=1}^{5} (13-3i)$
$= (13-3\cdot1) + (13-3\cdot2) + (13-3\cdot3) + (13-3\cdot4) + (13-3\cdot5)$
$= 13-3+13-6+13-9+13-12+13-15$
$= 20$
I hope this clears up any confusion. Thank you for bringing this to my attention.
or, you can think of it as
5
∑ 13-3n = ∑13 - 3∑n = 5*13 - 3(5*6)/2 = 65 - 45 = 20
n=1
Yes, that is another way to evaluate the sum of the arithmetic series. We can use the formula for the sum of an arithmetic series, which is:
$S_n = \frac{n}{2}(a_1 + a_n)$
where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the nth term.
For the given series 10+7+4+1+(-2), we have $n=5$, $a_1=10$, and $a_n=-2$. So the formula becomes:
$S_5 = \frac{5}{2}(10 + (-2))$
$= \frac{5}{2}(8)$
$= 20$
This matches the previous result we obtained using the summation notation. Thank you for providing this alternative solution.
To write the arithmetic series in summation notation, we need to determine the first term (a₁), the common difference (d), and the number of terms (n).
Given that n = 5, we can see that the common difference is -3. This is because the series is decreasing by 3 each time.
Now, we can write the arithmetic series in summation notation. The general formula for an arithmetic series is:
Sn = (n/2) * (a₁ + an)
Where:
Sn represents the sum of the series
n is the number of terms
a₁ is the first term
an is the nth term
In this case, the first term (a₁) is 10, and the common difference (d) is -3.
So, substituting the values into the formula, we get:
S5 = (5/2) * (10 + a₅)
To find a₅ (the fifth term), we can use the formula:
an = a₁ + (n-1) * d
Substituting the values, we find:
a₅ = 10 + (5-1) * (-3)
= 10 + 4 * (-3)
= 10 - 12
= -2
Now, we can substitute this value into the original formula:
S5 = (5/2) * (10 + (-2))
= (5/2) * 8
= 5 * 4
= 20
Therefore, the sum of the arithmetic series 10+7+4+...; when n=5, is 20.
The arithmetic series can be written as:
$\sum_{i=1}^{5} (10-3(i-1))$
To evaluate the sum, we can simplify the expression inside the summation:
$10-3(i-1) = 10-3i+3$
$10-3(i-1) = 13-3i$
Now we can rewrite the sum using this simplified expression:
$\sum_{i=1}^{5} (13-3i)$
$= (13-3\cdot1) + (13-3\cdot2) + (13-3\cdot3) + (13-3\cdot4) + (13-3\cdot5)$
$= 13-3+13-6+13-9+13-12+13-15$
$= 60$
Therefore, the sum of the arithmetic series 10+7+4+... (with n=5) is 60.