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Find an expression for the nth term of each sequence. Write down your expression in the form , where p, q . Justify your answers using technology.
Begin with the first sequence (log₂8, log₄8, log₈8, log₁₆8, log₃₂8, log₆₄8, log₁₂₈8) and determine an expression for the nth term:
The value 2 was used to determine the nth term.
1.
Apply the change of base formula that states:
Then apply the rule:
cancels out on both sides.
What remains is, where n represents the nth term in the sequence.
Here are two graphs to test validity:
y=
y=
Both graphs are identical towards each other, which indicates that both functions of the graphs are the same.
Next, consider the second sequence (log₃81, log₉81, log₂₇81, log₈₁81, log₂₄₃81, log₇₂₉81)
2.
Use the change of base formula
Then apply the rule:
cancels out on both sides.
What remains:
Here are two graphs to test its validity and accuracy:
y=
y=
Both graphs are identical towards each other, which indicates that both functions of the graph are the same.
Next, consider the third sequence (log₅25, log₂₅25, log₁₂₅25, log₆₂₅25, log₁₂₅25, , log₁₅₆₂₅25)
3.
Use the change of base formula:
Then apply the rule:
cancels out on both sides
What remains:
Here are two graphs to test its validity and accurateness:
Y=
Y=
Both graphs are identical towards each other, which indicates that both functions of the graph are the same.
4. Sequence expressed using the variables m, n, and k.
Use to change the base of the logarithm to ten.
cancels out.
What remains is. Derived from this, it can be concluded that the general expression for the nth term of each sequence in the form is .
Justification of this statement using technology:
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Now calculate the following, giving your answer in the form , where p, q, .
The answer was . This form will be used.
2.
3.
4.
Describe how to obtain the third answer in each row from the first two answers. Create two more examples that fit the pattern above.
1.
By recognizing that is the base in each of these logarithm, it is apparent that n=1 in the first logarithm and n=2 in the second logarithm. When added 2+3=5, therefore the third answer is obtained by multiplying the bases together in accordance with the rules of exponents.
The next two examples that would fit the pattern would therefore be:
2.
By recognizing is the base in each of these logarithms, it is apparent that n=1 is the first logarithm and n=2 in the second logarithm. When added 1+2=3, therefore the third answer is obtained by multiplying the bases together in accordance with the rules of exponents.
The next two examples that would fit the patter would therefore be:
3.
By recognizing is the base in each of these logarithms, it is apparent that n= -1
In the first logarithm n=-1 and in the second logarithm n= -3. When added n= -1 + -3= -4, therefore the third answer is obtained by multiplying the bases together in accordance with the rules of exponents.
The next two examples, which would fit the pattern would therefore be:
4.
In the first logarithm n=3 and in the second logarithm n=1. When added n=3+1=4, therefore the third answer is obtained by multiplying the bases together in accordance with the rules of exponents.
The next three examples that would fit the pattern would therefore be:
Let and . Find the general statement that expresses in terms of c and d.
and then find .
One law of logarithms states that:
Use the change of base formula:
then then
Take the Logarithm in base x:
Derived from it can be stated that:
Using the change of base formula the following expression is derived:
Substitute: and :
Multiply both sides by cd:
The general statement that expresses in terms of c and d is:
Test the validity of your general statement using other values of a, b, and x.
and
- Example: a=2, b=4, x=8
and
Check with the general statement:
General Statement Justified
- Example: a=5, b=125, x=25
and
Check with the general statement:
General Statement Justified
-
Example: a=1000, b=, x=10
and
Check with the general statement:
General statement justified.
Discuss the scope and/ or limitations of a, b, x.
The limitations of logarithms are, as previously stated:
a>0, a1, b>0
therefore the limitations for this question are as follows:
a>0, b>0, a≠1, b≠1, x>0
To check for validity of this statement:
Example: a=-2, b=2, x=4
and
It is impossible to have a negative power in this function, in this case a=-2. With these numbers:
, n>0.
The same applies for b, that a>0.
Example 2: a=10, b=1, x=100
and
n=0 as , which is impossible because it forced division to occur between that number and = error/ not possible
The example applies for a, that b≠1.
Example 3: a=4, b=8, c=-8.
and
As seen in example 1, it is impossible to have a negative power in one of these functions.
With these numbers: , k>0.
As a>0 and b>0, the product x should always be greater than 0, therefore x>0.
In summary:
a>0, b>0, a≠1, b≠1, x>0
Explain how you arrived at your general statement.
First, the sequences given with observed. After determining that each sequence had a constant exponent, but the bases did not remain constant, I used the change of base formula. This allowed the logarithm to be rewritten in terms of logs written with other bases.
then then
After observing the sequences and determining the nth term, the validity of the general formula was tested using other values of a, b, and x. The validity was proved through the formula:
After taking the logarithms in base x:
Therefore,
Then it is necessary to use the change of base formula once again to get the following expression:
Since variables were given, it was most likely that substiution would be needed to find the general formula:
Consequently, after multiplying both sides by cd, the general formula is:
