Find the sum of the geometric series 128 - 64 + 32 - . . . to 8 terms.
a) 85
b) 225
c) 86
d) 85/2
e) not possible
r = -64/128 = -1/2
a = 128
sum(8) = 128( 1 - (1/2)^8 / (1+1/2)
= 128(255/256)(2/3) = 85
With only 8 terms, it's easier for me to do the addition than remember the formula.
Adding in pairs, successive pairs are lower by a factor of 4.
64 + 16 + 4 + 1 = 85
128-64+32-16+8-4+2-1=64+16+4+1=80+5=85(!)
Whtav is the geometric sum of the first six terms in 256,128,64...
Well, let's see here. It seems like we have a geometric series with a first term of 128 and a common ratio of -1/2. To find the sum of a geometric series, we can use the formula:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio),
where n is the number of terms.
In this case, the first term is 128, the common ratio is -1/2, and we have 8 terms.
So let's plug in those values and see what we get.
Sum = (128 * (1 - (-1/2)^8)) / (1 - (-1/2))
Calculating that, we find:
Sum = (128 * (1 - 1/256)) / (3/2)
= (128 * (255/256)) / (3/2)
= (128 * 255) / (3 * 256)
= 32 * 255 / 64
= 8160 / 64
= 127.5
Oops! It seems that we have a decimal answer, but the options are all whole numbers. So, unfortunately, it's not possible to find the sum of this geometric series using just 8 terms. The correct answer is e) not possible.
Better luck with the next question!
To find the sum of a finite geometric series, you can use the formula:
S = a * (1 - r^n) / (1 - r)
Where:
S is the sum of the geometric series
a is the first term
r is the common ratio
n is the number of terms
In this case, the first term (a) is 128 and the common ratio (r) is -1/2 because each term is obtained by multiplying the previous term by -1/2. The number of terms (n) is 8.
Let's substitute these values into the formula:
S = 128 * (1 - (-1/2)^8) / (1 - (-1/2))
Simplifying the equation:
S = 128 * (1 - 1/256) / (3/2)
S = 128 * (255/256) / (3/2)
S = (128 * 255) / (256 * 3/2)
S = (128 * 255 * 2) / (256 * 3)
S = 32640 / 768
S = 42.5
Therefore, the sum of the geometric series to 8 terms is approximately 42.5.
None of the given answer choices are an exact match.