Im stuck on a lot of questions, here they are:
24 + 12 + 6 + ...(S7)
Sn = 24(2^7-1)/2-1
S7 = 24(127)/1
S7 = 3048/1
Thats what I got as an answer, but it's wrong, the real answer is a fraction answer, and its 381/8. How did they get that???
The other one was:
512 + (-256) + 128 +... +(-1)
Sn = -2(-1)-512/-2-1
= -510/-3
This answer is somehow wrong too:/
And the other one:
-8 -2 -1/2...-1/128
Sn = 4(-1/128)-8/4-1
Sn = 4/1 * 128/1 = 512 *-8
Sn = -4096/3
= This is also the wrong answer. Im soo confused ://// It's supposed to be 1356/128
1st:
your r value is 1/2, you used it as +2
Sum(7) = 24((1/2)^7 - 1)/(1/2-1)
= 24(1/128 - 1)/(-1/2)
= 24(-127/128)(-2) = 381/8
2nd:
a = 512 , r = -1/2
first find number of terms, last term is -1
t(n) = ar^(n-1)
-1 = 512(-1/2)^(n-1)
-1/512 = (-1/2)^(n-1)
(-1/2)^9 = (-1/2)^(n-1)
9 = n-1
n = 10
so Sum(10) = 512((-1/2)^10 - 1)/(-1/2-1)
= 512(-513/512)(-2/3)
= 342
3rd:
-8 - 2 - 1/2 - ... - 1/128
a = -8 , r = 1/4 , n = ??
term(n) = ar^(n-1)
-1/128 = -8(1/4)^(n-1)
1/1024 = (1/4)^n-1
(1/4)^5 = (1/4)^(n-1)
5 = n-1
n = 6
the series is
-8 -2 -1/2 -1/8 -1/32 - 1/128
sum(6) = -8((1/4)^6 - 1)/(1/4 -1)
= -8(-4095/4096)/(-3/4)
= -8(-4095/4096)(-4/3)
= -1365/128
Let's break down each question and see how to arrive at the correct answers step by step.
1. 24 + 12 + 6 + ...(S7)
The given series follows a pattern of dividing each term by 2. To find the sum of the series, we can use the formula for the sum of a geometric series: Sn = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is 24, the common ratio (r) is 1/2, and the number of terms (n) is 7.
Using the formula:
S7 = 24 * (1 - (1/2)^7) / (1 - 1/2)
= 24 * (1 - 1/128) / (1/2)
= 24 * (127/128) / (1/2)
= 24 * (127/128) * (2/1)
= 3048/1
It seems like you got the same answer so far. However, to simplify it further, we can write 3048/1 in fraction form by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 3048 and 1 is 1. Dividing both sides by 1 gives us:
S7 = 3048/1 = 3048.
So, the correct final answer is 3048.
2. 512 + (-256) + 128 + ... + (-1)
The given series alternates between positive and negative terms, starting with a positive term. One approach to finding the sum is to group the terms:
(512 + (-256)) + (128 + (-64)) +...+ (-1)
Each group of terms has a common ratio of -1/2. Here again, we can use the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r)
In this case, the first term of each group (a) is 512, the common ratio (r) is -1/2, and the number of groups (n) is 9 (since there are 9 terms in total).
Using the formula:
Sn = 512 * (1 - (-1/2)^9) / (1 - (-1/2))
= 512 * (1 - 1/512) / (1 + 1/2)
= 512 * (511/512) / (3/2)
= 512 * (511/512) * (2/3)
= 363,392/3
Again, we can simplify this answer by dividing the numerator and denominator by their GCD. In this case, the GCD of 363,392 and 3 is 1. Dividing both sides by 1 gives us:
Sn = 363,392/3
So, the correct final answer is 363,392/3.
3. -8 - 2 - 1/2...-1/128
The given series follows a pattern of dividing each term by 2. The sum can be found using the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r)
In this case, the first term (a) is -8, the common ratio (r) is 1/2, and the number of terms (n) is 8.
Using the formula:
Sn = -8 * (1 - (1/2)^8) / (1 - 1/2)
= -8 * (1 - 1/256) / (1/2)
= -8 * (255/256) / (1/2)
= -8 * (255/256) * (2/1)
= -8 * (255/256) * (2/1)
= -4096/3
Again, we can simplify this answer by dividing the numerator and denominator by their GCD. In this case, the GCD of -4096 and 3 is 1. Dividing both sides by 1 gives us:
Sn = -4096/3
So, the correct final answer is -4096/3.