1) 2)
Cleary, the pattern is noticeable as the sequence goes on. Therefore, I was able to find the term for each sequence, writing it in the form where . For the general logarithmic sequence, the term is the following. …,
That being the case, the nth term for the three examples of logarithmic sequences are the following: 1)
2)
3)
To justify my answer using technology, I used excel to verify that the pattern continues (see appendix 1). By carrying out the pattern to where n=50, I was able to confirm that the pattern continues.
Now in this investigation, I will analyze a new set of logarithmic sequences, to try and discover a new pattern that they all share in common. The new logarithmic sequences are the following:
1) 3)
2)
Using the first sequence, I was able to discover that when you multiply the bases of the first and second log, you get the base of the third log, for example: 4 x 8 = 32.
To evaluate the first logarithmic sequence more in depth to try and find a pattern I did the following:
The following equation where c is the denominator of the answer on the first log, and where d is the denominator of the answer of the second log within the sequence. While x is the third term. Thus c = 3 and d= 2. To make more sense, here is the equation:
To confirm that this pattern stays true, multiple other sequences will be used with the same formula: .
This is the next logarithmic sequence I will use to prove the pattern.
Therefore, .
Its noticeable that this pattern works for both logarithmic sequences, but to confirm that the pattern stays true, another logarithmic sequence will be used.
Therefore,
This pattern seems to work for the above examples of logarithmic sequences, but to prove that this pattern continues to work for all logarithmic sequences like the ones above then a general formula must be established.
If the following two statements are true,
Then this next statement must also be true,
Therefore, the statements above will be used to prove that within this next statement:
This investigation aimed to prove that this equation:
is true for all similar logarithmic sequences. How ever, this cannot be used for any log where the base of a times the base of b does not equal c. Also this equation does not work for negative numbers because you cannot take the log of a negative number.
In conclusion, I arrived at my general statement for both parts by using multiple logarithmic sequences to confirm that a pattern actually occurs, and is true for all similar logarithmic sequence.
Appendix 1: shows the continuation of pattern within Part one of IA.
