12. What is the sum of the finite arithmetic series?
(–5) + 0 + 5 + 10 + ... + 65
answer options
900
455
450
445
the common difference is 5
there are 15 terms
[65 + (-5)] * 15/2 = ?
65+ (-5)=60
15/2=7.5
7.5*60=450
okay, i get it now, thanks!
Well, let's find out together, shall we?
To find the sum of a finite arithmetic series, we can use the formula Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term is -5 and the last term is 65. Let's try to figure out the number of terms (n) first. We can use the formula: l = a + (n - 1)d, where d is the common difference.
By plugging in the values we know, we get: 65 = -5 + (n - 1)d.
Now the fun part: try different values for the common difference (d) until we find one that satisfies the equation! May I suggest starting with d = 5, since we have a series that increments by 5 each time?
Using d = 5, we get: 65 = -5 + (n - 1)5. Simplifying this equation, we get: 65 = -5 + 5n - 5. Combining like terms, we find: 65 = 5n - 10. Add 10 to both sides: 75 = 5n. Divide both sides by 5: n = 15.
So, it seems like our series has 15 terms. Now let's plug this into the sum formula:
Sn = (15/2)(-5 + 65) = (15/2)(60) = 450.
Drumroll, please! The sum of the series is 450. Ta-da!
But hey, don't let my nerdy math routine take away the humor – why did the arithmetic series go to the circus? Because it heard there were lots of "sum-ring" acts!
To find the sum of a finite arithmetic series, you can use the formula:
Sn = (n/2) * (a1 + an)
where:
- Sn is the sum of the series
- n is the number of terms
- a1 is the first term
- an is the last term
In this case, the first term (a1) is -5, and the common difference between each term is 5. We need to find the last term (an).
To find the last term (an):
an = a1 + (n - 1)d
where:
- an is the last term
- a1 is the first term
- n is the number of terms
- d is the common difference
In this case, a1 = -5, d = 5, and we need to find n.
To find n:
an = 65
-5 + (n - 1) * 5 = 65
5(n - 1) = 70
n - 1 = 14
n = 15
Now, with the value of n, we can find the sum (Sn):
Sn = (n/2) * (a1 + an)
Sn = (15/2) * (-5 + 65)
Sn = (15/2) * 60
Sn = 450
Therefore, the sum of the arithmetic series is 450.