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

✏️📓 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 ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
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 #80

Equation of a Circle. We're definitely keeping our algebra skills sharp with this week's problem. 🧠 As the saying goes, "Repetition is the mother of all learning". So, you will definitely see skills being repeated in the problems as well the combination of them to solve the problems. The equation of a circle is part of a larger topic called conic sections. Let's  go!  🏃🏿‍♀️

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

       ✔️     Substitution
       ✔️     Completing the square technique
       ✔️     How to solve a quadratic equation

            

WMP #80 wants us to...

Happy solving!

Check back on Saturday, April 9th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Part a.

To determine the center and radius of the circle, the completing the square technique needs to be used on the given equation. To start that process, the constant needs to be isolated. Once the equation is in standard form, then the center and radius can be identified.

The center of the circle is (1, -1) and its radius is 1 unit. This was figured out without graphing it. We'll view the graph later in this post.

Part b.

To show that the point (1, -2) is a point of intersection for circle C and the given line, the substitution method was used. The equation of the line was substituted for y in the equation of the circle. The equation of the circle was then simplified into a factorable quadratic equation. This allows for values of x to be solved for...one of which is the x-value in the given point. It is then shown that the corresponding y-value is indeed the y-value in the given point.

Now that we've completed both parts of the problem, let's view the graph:

**This plot was created using Geogebra's Graphing Calculator. Click image to enlarge.

 

▪️ 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. ✏️
Let's move on to WMP! #81. 💪🏿

Cheers!

The Younge Lady