Calculate the sum of the infinite series 72 + 60 + 50 + 125/3 + ...
489
432
360
307
Can someone please help me with this one? I can never solve these on my own:(
Nah. Where did you get the 40? 50 * 5/6 = 250/6 = 125/3
so, you have a genuine GP, with
a = 72
r = 5/6
So, S = 72/(1 - 5/6) = 72*6 = 432
Oh ok, thank you so much.
Well, solving an infinite series is like trying to catch a cloud; it's never-ending! But don't worry, I'm here to bring some laughter into the equation.
Now, let's take a closer look at the series. We can see that each term is getting smaller and smaller. It's like a game of limbo, how low can you go?
So, I'll do you a favor and sum up the first few terms for you. We have 72 + 60 + 50 + 125/3, which... uh-oh, I seem to have misplaced my calculator.
But hey, who needs numbers when you've got jokes? What do you call friends who love math? Algebros! They'll add up the laughter, even if the infinite series gives you a headache.
As for the answer, let's go with 360. But don't let my clown antics fool you – always double-check with a qualified mathematician to be sure!
To calculate the sum of an infinite series, we need to determine if the series is convergent or divergent. In this case, we'll analyze the series and see if we can identify a pattern.
Looking at the terms of the series, we can observe that each term is decreasing compared to the previous one. However, it's not immediately clear what the pattern is. Let's try calculating the ratio of consecutive terms:
First term: 72
Second term: 60
Ratio: 60/72 = 5/6 ≈ 0.833
Second term: 60
Third term: 50
Ratio: 50/60 = 5/6 ≈ 0.833
Third term: 50
Fourth term: 125/3
Ratio: (125/3)/50 ≈ 8.333/6 ≈ 1.388
We can see that the ratio between consecutive terms is not constant. Therefore, this series doesn't seem to have a geometric progression, which is usually easier to deal with.
Given this, we need to try a different approach. One possibility is to group the terms in a way that simplifies the calculation. Let's group the terms as follows:
(72 + 60) + (50 + 125/3) + ...
Now, we can observe that the first two terms can be simplified to 132:
(132) + (50 + 125/3) + ...
Continuing this pattern, we see that each group of two terms can be simplified to 132. Hence, the series can be rewritten as:
132 + 132 + 132 + ...
Now, we have an infinite series of identical terms, which is much easier to deal with.
To calculate the sum of this new series, we'll use the formula for the sum of an infinite geometric series:
sum = a / (1 - r)
where:
a = the first term of the series
r = the common ratio (ratio between consecutive terms)
In our case, a = 132 and r = 1 (since all terms are the same).
Plugging these values into the formula, we get:
sum = 132 / (1 - 1) = 132 / 0
We have an indeterminate form since division by zero is undefined. Therefore, this series is divergent, meaning it doesn't have a finite sum.
So, none of the given answer choices (489, 432, 360, 307) are correct.
if it is geometric, r=60/72=10/12=5/6
adn then 5/6(60=50
and then 40(5/6)=200/6=100/3, so that is not a geometric series. Did your teacher make an error? Lets test if she meant it to be a geometric series, r=5/6
sum=72/(1-5/6)=72*6=amazing, the answer is there.
So I bet your teacher meant 100/3 as the fourth term.