the question is find the sum of the first eight terms in the series
3-12+48-192+... is
1)-13,107
2)-21,845
3)-39,321
4)-65,535
I got that a1 is 3 and a8 was -49152. I wanted to make sure this was a geometric sequence with a r of -4 and tat the + didn't mean to add. I got that the answer was 3, -39,321. THANK YOU
S8 = a(r^8-1)/(r-1) = 3((-4)^8-1)/(-4-1) = -39321
looks good
Well, if the series is indeed a geometric sequence with a common ratio of -4, then we can use the formula for the sum of a geometric series to find the answer. The formula is given by:
S = a * (1 - r^n) / (1 - r)
Where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the given values, we have a = 3, r = -4, and n = 8:
S = 3 * (1 - (-4)^8) / (1 - (-4))
= 3 * (1 - 65536) / (1 + 4)
= 3 * (-65535) / 5
= -393,321 / 5
= -78,664.2
According to my calculations, the sum of the first eight terms in the series is approximately -78,664.2. So, none of the answer options provided is correct. But hey, at least now you know the correct sum, and with a bonus of some mathematics humor!
To determine if the series is a geometric sequence, we need to check if the ratio between consecutive terms is constant.
Given the terms:
a1 = 3
a2 = -12
We can calculate the ratio, r, by dividing a2 by a1:
r = a2 / a1 = (-12) / 3 = -4
Since the ratio between consecutive terms is constant, with r = -4, we can conclude that the series is indeed a geometric sequence.
To find the sum of the first eight terms in a geometric sequence, we use the formula for the sum of a finite geometric series:
Sn = a * (r^n - 1) / (r - 1)
Where:
Sn = sum of the first n terms
a = first term
r = common ratio
n = number of terms
Using the given values, we have:
a = 3
r = -4
n = 8
Sn = 3 * (-4^8 - 1) / (-4 - 1)
= 3 * (65536 - 1) / (-5)
= 3 * 65535 / -5
= -3 * 13107
= -39,321
Therefore, the sum of the first eight terms in the series 3-12+48-192+... is -39,321 (option 3).
To find the sum of the first eight terms in the given series, you are correct in identifying that it is a geometric sequence with a common ratio (r) of -4.
To verify this, we can divide each term by the previous term to check if the ratio is constant. Let's check the first few terms:
a2 / a1 = (-12) / 3 = -4
a3 / a2 = (48) / (-12) = -4
a4 / a3 = (-192) / 48 = -4
As the ratio is consistent at -4, we can conclude that it is a geometric sequence.
Now, to calculate the sum of the first eight terms, we can use the formula for the sum of a geometric series:
Sn = a1 * (1 - r^n) / (1 - r)
Where:
- Sn is the sum of the first n terms,
- a1 is the first term,
- r is the common ratio, and
- n is the number of terms.
In this case, a1 = 3, r = -4, and n = 8.
Plugging these values into the formula, we get:
S8 = 3 * (1 - (-4)^8) / (1 - (-4))
= 3 * (1 - 65536) / (1 + 4)
= 3 * (-65535) / 5
= -196605 / 5
= -39321
Therefore, the sum of the first eight terms in the series 3-12+48-192+... is -39,321.
Option 3, -39,321, is the correct answer.