|
Question 472811: the sum of the first 5 terms of the geometric sequence 1/24, -1/12, 1/6, -1/3... is 5/72 ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ratio appears to be -2
1/24 * -2 = -1/12
-1/12 * -2 = 1/6
1/6 * -2 = -1/3, etc.
the formula for a geometric series with n terms is:
S[n] = a * (1-r^n)/(1-r)
a is the first term in the sequence.
r is the common ratio.
for example:
assume a is equal to 10 and r is equal to 5 and n is equal to 2.
s[2] = a * (1-r^2)/(1-r) = 10 * (1-5^2)/(1-5) = 60
it's easy to see that the sum for this sequence is equal to 60 because:
a = 10
a*r = 10*5 = 50
sum of a + a*r = 60
applying this to your problem, this formula should provide us with the correct answer.
in your problem:
a = 1/24
r = -2
n = 5
the formula becomes:
s[5] = 1/24 * (1-(-2)^5)/(1-(-2))
this becomes:
s[5] = 1/24 * (33)/(3) = 1/24 * 11 = 11/24
we can see if this is correct by simply adding up the terms.
a = 1/24 (first term)
1/24 * (-2) = -1/12 (second term)
-1/12 * (-2) = 1/6 (third term)
1/6 * (-2) = -1/3 (fourth term)
-1/3 * (-2) = 2/3 (fifth term)
add these terms up together and you get:
1/24 - 1/12 + 1/6 - 1/3 + 2/3
convert everything to a fraction with a denominator of 24 to get:
1/24 - 2/24 + 4/24 - 8/24 + 16/24
this becomes:
(1 - 2 + 4 - 8 + 16) / 24 which becomes:
(21 - 10) / 24 which becomes:
11/24.
the formula works.
the answer is 11/24.
|
|
|
| |
