How many terms of the series 25+19+13 +… are required to make the sum −20?
Well, let me grab my calculator... just kidding! I don't need one. Let's do some funny math instead.
So, we have a series that starts at 25 and decreases by 6 each time. To find the number of terms needed to reach a sum of -20, let's imagine each term as a step on a staircase. We take these steps downwards until we hit -20.
If we step on a banana peel on the way and slip by accident, we can always pretend like we did it on purpose. Because, you know, life is all about finding balance... or at least making it look like we meant to trip.
Anyway, the formula for the sum of an arithmetic series is S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
By plugging in the values, we get -20 = (n/2)(25 + l). Since we're trying to find the number of terms (n), let's rearrange the equation a bit.
-20 = (25 + l)(n/2)
-40 = 25 + l
l = -65
Now that we know the last term (l), we can find the number of terms (n) by substituting it into the equation we got earlier.
-20 = (n/2)(25 - 65)
-20 = (n/2)(-40)
n = 1
Well, I guess one term is enough for this series to make a sum of -20. Looks like we didn't need that banana peel after all. But hey, at least we had a good laugh along the way!
To find the number of terms required to make the sum -20, we need to find the pattern of the series.
The given series is formed by subtracting 6 from the previous term: 25, 19, 13, ...
To find the general term of the series, we can use the formula:
aₙ = a₁ + (n-1)d
where a₁ is the first term, d is the common difference, and n is the term number.
In this case, a₁ = 25 and d = -6, since we are subtracting 6 from each term.
Using the formula, we can find the general term:
aₙ = 25 + (n-1)(-6)
Next, we need to find the sum of the series up to the nth term. The sum of an arithmetic series is given by the formula:
Sₙ = (n/2)(a₁ + aₙ)
where Sₙ is the sum of the series.
In this case, we want to find the number of terms required to make the sum -20, so we can set up the equation:
-20 = (n/2)(25 + aₙ)
Since we don't know the value of aₙ, we need to find its expression in terms of n. By substituting the expression for aₙ from earlier into the equation, we can solve for n:
-20 = (n/2)(25 + 25 - 6n)
-20 = (n/2)(50 - 6n)
Simplifying the equation:
-20 = (n/2)(50 - 6n)
-40 = n(50 - 6n)
-40 = 50n - 6n²
Rearranging the equation:
6n² - 50n - 40 = 0
Now we can solve for n by factoring or using the quadratic formula.
Using the quadratic formula:
n = (-b ± √(b² - 4ac)) / 2a
In this case, a = 6, b = -50, and c = -40.
n = (-(-50) ± √((-50)² - 4(6)(-40))) / (2(6))
n = (50 ± √(2500 + 960)) / 12)
n = (50 ± √(3460)) / 12)
n = (50 ± 58.87) / 12)
n₁ = (50 + 58.87) / 12 ≈ 8.24
n₂ = (50 - 58.87) / 12 ≈ -0.72
Since n cannot be negative, we take the positive value, n ≈ 8.24.
Therefore, we need approximately 8 terms of the series to make the sum -20.
To find out how many terms of the series are required to make the sum -20, we need to find the pattern in the series and then solve for the value of n (the number of terms).
In the given series: 25 + 19 + 13 + ...
To find the pattern, we can observe that each term is obtained by subtracting 6 from the previous term:
25 - 6 = 19
19 - 6 = 13
13 - 6 = 7
So, the common difference in this arithmetic sequence is -6.
Now, let's find the formula for the nth term of the series:
An = 25 + (n-1)(-6)
To find the sum of n terms, we can use the formula:
Sn = (n/2)(2A1 + (n-1)d)
In this case, we know that the sum (Sn) is -20. So, we can write the equation as:
-20 = (n/2)(2(25) + (n-1)(-6))
Now, we can solve the equation for n to find out how many terms are required to make the sum -20.
a = 25
d = -6
You want n such that
n/2 (25 + (n-1)(-6)) = -20
Now just solve for n