how many terms of the series -8,-6-,-4.... make the sum 90?
a=-8
d=+2
n=?
S(n) = n/2[2a+(n-1)d]
90 = n/2[-16 + 2(n-1)]
this works out to
n^2 - 9n - 90 = 0
this factors nicely and has two answers, one of which will make no sense.
Let me know what you got.
To be smart in math
To be smart in matj.
Ah, the ol' "sum of a series" conundrum. Let's solve this together, shall we?
First, let's find the common difference between the terms. By subtracting each term from the previous one, we can see that the common difference is 2.
Now, let's find the number of terms, shall we?
We can use the formula for the sum of an arithmetic series (Sn = (n/2)(2a + (n-1)d)), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference.
Plugging in the values, we have 90 = (n/2)(2(-8) + (n-1)2).
Simplifying this equation, we get 90 = (n/2)(-16 + 2n - 2).
Now, let's lighten things up a bit. Why did the scarecrow win an award? Because he was outstanding in his field!
Back to the equation. Let's continue simplifying: 90 = (n/2)(-18 + 2n).
Expanding this further, we have 90 = -9n + n².
Rearranging the equation, we get n² - 9n + 90 = 0.
Now, let's find the values of n. Using the quadratic formula, we have n = (-(-9) ± √((-9)² - 4(1)(90))) / (2(1)).
Simplifying this, we have n = (9 ± √(81 - 360)) / 2.
Continuing to simplify, we have n = (9 ± √(-279)) / 2.
Uh-oh! The square root of a negative number! That's not possible within the realm of real numbers. So, it seems that there are no *real* solutions for this problem of ours.
But fear not, my friend! There's always a solution hidden somewhere. In this case, it seems that there are no terms in the series that will make the sum equal to 90. So, you can pack your bags and head home empty-handed.
Just remember, the journey is often more important than the destination!
To find out how many terms of the series -8, -6, -4... make the sum 90, we need to analyze the given sequence.
It appears that the given sequence is an arithmetic progression, where each term is obtained by adding 2 to the previous number. The general formula for the nth term of an arithmetic progression is given by:
an = a1 + (n - 1) * d
In this case, a1 is the first term of the sequence (-8), n is the number of terms we want to find, and d is the common difference between the terms (2).
Let's set up the equation to find the number of terms that sum up to 90:
90 = (-8) + (-8 + 2) + (-8 + 2*2) + ... + [a1 + (n-1)d]
Now we can simplify the equation:
90 = -8 - 6 - 4 - ... - 8 + (n-1) * 2
Next, let's express the sum of the first (n-1) terms of the arithmetic series as:
Sn = (n/2) * (a1 + an)
Substituting the known values for Sn and an, we have:
90 = (n/2) * (-8 + [a1 + (n-1)*2])
Simplifying further:
90 = (n/2) * (-8 + (-8 + 2n - 2))
90 = (n/2) * (2n - 16)
Factoring out 2:
90 = (n/2) * 2 * (n - 8)
90 = n * (n - 8)
Rearranging the equation:
n^2 - 8n - 90 = 0
Now, we have a quadratic equation. To solve for n, we can factor, use the quadratic formula, or complete the square. Let's use factoring:
(n - 10)(n + 9) = 0
The values of n that satisfy the equation are n = 10 and n = -9. However, since the number of terms cannot be negative, we discard the negative value.
Therefore, the number of terms that make the sum equal to 90 is 10.