Weekly Math Problem #82

Geometric Series. This week's problem is a second part to last week's problem, WMP #81. Since we looked at a geometric sequence last week, why don't we look the corresponding geometric series this week? Well...we are. ✏️ 👍🏿

To solve this week's problem in completion, you need to recall the following math skills and information:

       ✔️     How to find the sum of the first set of numbers from a geometric sequence  

            

WMP #82 says to...

Happy solving!

Check back on Saturday, April 30th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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And we're back! Of course, to find the sum of the first eight terms of our (or any) geometric series, we can write it down and find the sum manually...but...what if, instead of eight terms, we had had to find eighteen terms?! I don't want to write that many terms. And guess, what? We don't have to. There's a formula for that!

Here is the formula:

In our situation, we have the following:

Well, it looks like the sum of the first eight terms of the geometric series whose first term is 2 with a ratio of 3 is 6560.

Now, just for fun, let's find the sum manually to verify that we've used the formula correctly. (This will also help us to appreciate the formula more.) Here are the first eight numbers generated from last week's geometric sequence with its corresponding sum:

...And it's a match! 

▪️ Did you find this problem easy? Hard? Somewhere between easy and hard?

▪️ Leave a comment to let me know what you thought about this week's problem. 

Thank you for solving with me this week. ✏️
WMP! #83 is up next. 🤓

Cheers!

The Younge Lady

 

Weekly Math Problem #81

Geometric Sequence. It has been a loooong while since I have done any work with geometric sequences...or series. So, now is as good a time as any to do it. 🤷🏿‍♀️ ✏️

To solve this week's problem in completion, you need to recall the following math skills and information:

       ✔️     Understanding what a geometric sequence is
       ✔️     How to find the nth term in a geometric sequence

            

WMP #81 says to...

Happy solving!

Check back on Saturday, April 16th for the solution, which will be posted below ⬇️.

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Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Alright folks... I don't know about you, but I needed a way to organize the information; I need to see what I'm working with. So, I used a chart:

The chart shows what information I have, what information is missing, and helps me to see how I can approach the problem. From the chart, I can see that there are four factors (or ratios) between the second and the sixth terms. I can also see that I have enough information to figure out what r is.

Now that I know the common ratio is 3, I can use that information to find the first term. Why do I need to find the first term? Well...in the formula needed to find the n-th term, the first term is required. So, let's proceed to see what the first term is. That is done by taking the second term and dividing it by the common ratio of 3.

 

Cool...the first term, a1, is 2. It looks like I have enough information to find the 8th term. But...before I do that, I want to verify that the information I have--the common ratio of 3 and the first term, a1, of 2 will generate the 6th term. 

Nice! It works. With confidence, the problem can be completed. The 8th term can be found.

There you have it. The 8th term of a geometric sequence whose 2nd and 6th terms are 6 and 486, respectively, is 4,347.

▪️ Did you use the same methods above to solve the problem? If so, please share.

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. ✏️
WMP! #82 is up next. 🤓

Cheers!

The Younge Lady

Weekly Math Problem! #34

Inverse of a Function. I have different ways of choosing problems to solve on this blog. Sometimes I go through books and notes I have. Other times, I use Google. 😁 This week, Google won as my method due to a lack of time. 🤷🏿‍♀️ What can I say? So, I decided to check out some precal topics and found a lighter problem to use. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the inverse of a function

That's it? Yup...that's it. WMP! #34 wants us to...

Happy solving!

Check back on Friday, January 22nd for the solution, which will be posted below ⬇️.

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Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Let's go right on ahead and jump in this thing! 

Here comes the fun 🤪 part! (At least I think it's fun. 🤷🏿 ) The equation right above here is a literal equation. We are going to solve it but, the answer we get will not be a number. By placing x over 1, it can be more clearly seen that the above equation is a proportion.

We're just about done. Just one more thing...a mere formality:

That's it! We've found the inverse function, f -1. 🤸🏿‍♀️ Yay! Like the original function, the inverse is also a rational function, with a variable present in the denominator. That means we cannot use, or plug-in, any value for x that we like. To continue with the second part of the question, we will find the domain for f -1 by identifying which value of x will cause the function to be undefined. I did something slightly unconventional to find the value. Instead of setting the denominator equal to zero, I set the denominator not equal zero and proceeded normally:

Now, I can write the domain formally. I don't really have a preference for notation, so I did it two ways:

 

You know I wouldn't end this blog post without graphing the functions, right? Okay, great! (I tried something a little different with the color scheme. What do you think?) When a function and its inverse are plotted on the same set of axes, they will be mirror images of each other. The mirror is the line y = x.

**This plot was created using Geogebra's Graphing Gaclulator.

▪️ Let me know what you thought about this week's problem down below.

Thanks for solving with me this week!

Up next...WMP! #35

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Cheers!

The Younge Lady

Weekly Math Problem! #30

De Moivre's Theorem. Sometimes, when I'm thinking about what problem to post for the week, topics just come to mind. Somehow, DeMoivre and his theorem was this week's winner. 🎉 The last time I really encountered this topic was well over five years ago...way back when I took a precalculus course. Now that I think about it, 🤔 I don't recall going over this topic in any tutoring sessions over the years. But then again, I've been tutoring for a long time and may not really remember if I actually did or not. 🤷🏿

I would also like to say the following: When I learned this topic, I was enrolled in a summer school course--about four weeks long. I learned the topic quickly without knowing what the practical applications of the theorem is. It felt like the topic was being learned for the sake of learning it, without any real-world connections being made. Even though I didn't ask the question then, I can ask it now and find an answer somewhere on this internet. Okay, I am done with that small rant. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to convert complex numbers to polar form
        ✔️     How to apply DeMoivre's Theorem to find powers of complex numbers
        ✔️     How to evaluate trigonometric functions

WMP! #30 wants us to...

Happy solving!

Check back on Friday, December 4th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Lowkey, I was a little annoyed by the question, after I began solving it. 😒 I didn't realize the numbers were gonna be this crazy. 😅  As soon as I saw 520 ...👀...smh...no words. I realized it was too late to turn back now, so I pushed forward to finish. I will say this, I was getting nervous about whether or not I was correct and needed to verify my numbers. Oh the anxiety I was having. 

Before applying De Moivre's Theorem, some preliminary calculations need to be performed.

Notice, I rounded the angle, 𝛳, in degrees to the nearest thousandth. I didn't want use radians because I am more comfortable with degrees. Next up, the application of De Moivre's Theorem.

Do you see what I mean about the numbers being crazy? 🤪 ...in the trillions! I am aware that answer above is not as accurate as it should be, due to my rounding of 𝛳 to the nearest thousandth. So, I have prepared a more accurate calculation below. 

My final thought: I didn't find the problem difficult to execute. Anxiety was triggered in me when I saw the first large number in my calculation. The anxiety lead me to begin doubting my mathematical abilities. Even though I am good at math, I still very much understand, what it feels like to be fearful when doing it.

▪️ What kind of emotions do you feel when doing math??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #31

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Cheers!

The Younge Lady