Can You Solve This Tricky Geometric Sequence Problem?

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In summary, we have a geometric sequence where the sum of the first three terms is 7 and the sum of the cubes of the first three terms is 73. After simplifying the calculation and using the hint to call the second term "x", we find that the common ratio of the sequence must be 2. The sequence can be written as 1, 2, and 4, with the first term being x/r and the third term being xr. There is no systematic way to find the answer, but it can be deduced that r must be 2 by looking at the values of r4 and 73/7. However, the difficult part is writing the steps to reach this solution. Writing a program to
  • #1
midododo11
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In a geometric sequence, the sum of the first three terms is 7
and the sum of the cubes of the first three terms is 73
find the sequence and how did you get it​
 
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  • #2
welcome to pf!

hi midododo11! welcome to pf! :smile:

hmm :rolleyes: … let's try to simplify the calculation by making it as symmetric as possible, sooo …

hint: call the second term of the sequence "x" :wink:​
 
  • #3


tiny-tim said:
hi midododo11! welcome to pf! :smile:

hmm :rolleyes: … let's try to simplify the calculation by making it as symmetric as possible, sooo …

hint: call the second term of the sequence "x" :wink:​

Hi tiny-tim!
I tried a lot and I always get stuck at the last step, I know the answer but I got it by trial of numbers
Take a look
[URL]http://latex.codecogs.com/gif.latex?\large&space;\\a=first&space;\term&space;\\r=&space;common\ratio&space;\\a+ar+ar^2=7&space;\&space;Equation&space;(1)&space;\\a^3+a^3r^3+a^3r^6=73&space;\&space;Equation&space;(2)&space;\\by\&space;dividing\&space;1&space;/&space;2&space;\\\frac{a(1+r+r^2)}{a^3(1+r^3+r^6)}=\frac{7}{73}&space;\\\frac{1+r+r^2}{a^2(1+r^3+r^6)}=\frac{7}{73}
[/URL]
 
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  • #4
don't use a = first, use x = second
 
  • #5
Sorry but I don't see the difference if used x as a second term the first term will be x/r and the third xr
 
  • #6
1 2 4
 
  • #7
midododo11 said:
Sorry but I don't see the difference if used x as a second term the first term will be x/r and the third xr

i've no idea why, but I've been misreading "geometric" as "arithmetic" …

maybe i was put off by the colour? :redface:

i don't think there's any systematic way of finding the answer, except possibly to say that r4 must obviously be a bit more than 73/7 … so r must be 2!​
 
  • #8
tiny-tim said:
i've no idea why, but I've been misreading "geometric" as "arithmetic" …

maybe i was put off by the colour? :redface:

i don't think there's any systematic way of finding the answer, except possibly to say that r4 must obviously be a bit more than 73/7 … so r must be 2!​

Ok, thanks
 
  • #9
I got 1,4, and 2...?
 
  • #10
ILoveScience said:
I got 1,4, and 2...?
yeah but it's required to write the steps that's the difficult part
 
  • #11
midododo11 said:
yeah but it's required to write the steps that's the difficult part

does writing the program to calculate it by brute force count? :P
 

FAQ: Can You Solve This Tricky Geometric Sequence Problem?

1. How do you solve a difficult sequence problem?

The first step in solving a difficult sequence problem is to carefully examine the given sequence and try to identify any patterns or relationships between the numbers. It can also be helpful to write out the sequence in different ways, such as in a table or graph, to better visualize the pattern. From there, you can use mathematical operations or other strategies to find the missing or next term in the sequence.

2. What are some common types of difficult sequence problems?

Difficult sequence problems can vary greatly in their complexity, but some common types include arithmetic sequences, geometric sequences, and Fibonacci sequences. Other types may involve more complicated patterns or require a combination of different types of sequences to solve.

3. Can difficult sequence problems have multiple solutions?

Yes, some difficult sequence problems may have multiple solutions. This can occur when there are multiple possible patterns or relationships within the given sequence. However, there are also cases where there is only one correct solution.

4. Is there a specific method or formula for solving difficult sequence problems?

There is no one-size-fits-all method or formula for solving difficult sequence problems. It often requires a combination of critical thinking, pattern recognition, and mathematical skills. Different strategies may work better for different types of sequences, so it's important to approach each problem with an open mind and be willing to try different approaches.

5. How can I improve my skills in solving difficult sequence problems?

Practice is the key to improving your skills in solving difficult sequence problems. The more you encounter and solve different types of sequences, the better you will become at recognizing patterns and applying mathematical concepts. It can also be helpful to study different strategies and techniques for solving sequence problems and to work with others to discuss and solve problems together.

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