The 3rd term of a geometric series is 18 , and the 5the term is 162 .Determine the sum of the of the first 7 terms ,where r < 0
162 = 18r^2
r = 3
18 = 9a, so a = 2
S7 = 2(1-3^7)/(1-3) = 2186
check:
2+6+18+54+162+486+1458 = 2186
Thanks for helping me
Yes
Everything is right, but
r = -3
not r = 3
because the problem stated that r < 0
Well, well, well, looks like we've got ourselves a geometric series problem! And it's not just any geometric series problem, it's a negative r kind of problem. Let's dive in, shall we?
We know that the 3rd term of the series is 18. So, let's call the first term "a" and the common ratio "r". Using the formula for the nth term of a geometric series, we can write the third term as:
a * r² = 18
Moving on to the 5th term, which is 162, we have:
a * r⁴ = 162
Now, let's do some detective work. Divide the equation for the 5th term by the equation for the 3rd term:
(a * r⁴) / (a * r²) = 162 / 18
Simplifying this, we get:
r² = 9
Now, since r < 0, we can jump to the conclusion that r = -3. Or we could use complex numbers and say r = 3i and make things really interesting, but that's a whole different story.
Anyway, now that we know r = -3, we can substitute it back into one of the original equations to find the value of a. Let's pick the equation for the 3rd term:
a * (-3)² = 18
a * 9 = 18
Solving for a, we find that a = 2.
Now, we have a = 2 and r = -3, we can find the sum of the first 7 terms of the series using the formula:
Sn = a * (1 - rⁿ) / (1 - r)
Plugging in the values, we have:
S7 = 2 * (1 - (-3)⁷) / (1 - (-3))
Simplifying this monstrous expression, we finally arrive at:
S7 = 2 * (1 - 2187) / 4
S7 = 2 * (-2186) / 4
And the grand finale...
S7 = -2186 / 2
S7 = -1093
So, the sum of the first 7 terms of the geometric series, where r < 0, is a big ol' -1093. Isn't math just a barrel of laughs?
To solve this problem, we need to first find the common ratio (r) of the geometric series. Once we know the common ratio, we can use the formula for the sum of a geometric series to find the sum of the first 7 terms.
Let's start by finding the common ratio (r):
We are given that the 3rd term is 18 and the 5th term is 162.
The formula to find the nth term of a geometric series is:
tn = a * (r ^ (n-1))
Where:
tn = nth term
a = first term
r = common ratio
n = term number
We can use the formula to find the values of tn for the 3rd and 5th terms:
18 = a * (r ^ 2) ...(1)
162 = a * (r ^ 4) ...(2)
Dividing equation (2) by equation (1), we get:
162 / 18 = (a * (r ^ 4)) / (a * (r ^ 2))
9 = r^2
Taking the square root of both sides, we find:
r = ±3
Given that r < 0, we can conclude that r = -3.
Now that we know the common ratio, we can find the sum of the first 7 terms (S7) using the formula for the sum of a geometric series:
S7 = a * ((1 - r^7) / (1 - r))
Substituting the values we have:
S7 = a * ((1 - (-3)^7) / (1 - (-3)))
Simplifying further:
S7 = a * ((1 - 2187) / (1 + 3))
S7 = a * ((-2186) / 4)
To find the value of 'a', we can use one of the terms of the series. Let's use the 3rd term, which is 18:
18 = a * (-3)^(3-1)
18 = a * (-3)^2
18 = a * 9
Dividing both sides by 9, we get:
a = 2
Now we can substitute the value of 'a' back into the formula for the sum:
S7 = 2 * ((-2186) / 4)
Simplifying further:
S7 = -1093