You passed this Milestone
21 questions were answered correctly.
1 question was answered incorrectly.
1
Find the sum of the first 10 terms of the following geometric sequences:
RATIONALE
This is the formula to find
...
You passed this Milestone
21 questions were answered correctly.
1 question was answered incorrectly.
1
Find the sum of the first 10 terms of the following geometric sequences:
RATIONALE
This is the formula to find the sum of a finite geometric sequence. We will use
information from the given sequence to find values for , , and .
Let's start by finding , the value of the first term.
In the above sequence, the first term is . So we will substitute for
in the sum of a geometric sequence formula. Next, let's determine the
variable .
To find , divide the value of any term by the value of the term before it to find the
a subscript 1 r n
a subscript 1
a subscript 1
r
r
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common difference. For example, so . Finally, we need to review how
many terms we want to consider, which will be .
We are asked to find the sum of the first 10 terms, so . We can now
substitute values in for , , and to solve for .
Once the values for , , and have been plugged into the sum
formula, we can simplify the numerator.
to the power of is . Next, evaluate the subtraction in both the
numerator and denominator.
minus is and minus is . Then, divide the numerator and
denominator.
The negative values in the numerator and denominator cancel to result in a positive
value of . Finally, multiply this by to find the sum.
The sum of the first ten terms in the sequence is .
CONCEPT
Sum of a Finite Geometric Sequence
2
Suppose we have two functions: and .
Find the value of .
RATIONALE
n
n equals 10
a subscript 1 r n S subscript n
a subscript 1 r n
2 10 1024
1 1
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To evaluate , evaluate and separately, and then multiply the
values together. Start by evaluating first by substituting for in the given
function.
is plugged in for all instances of in the function . Next, evaluate the
exponent.
squared is . Then, add and together.
plus is . Repeat this process for by substituting for in the given
function.
is plugged in for all instances of in the function . Next, evaluate the
multiplication.
times is . Then, add and together.
plus is . Finally multiply the results of and together.
We know that and . Multiply these values together.
times is .
CONCEPT
Multiplying and Dividing Functions
3
Evaluate the following expression using the properties of logarithms.
RATIONALE
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To evaluate this expression, recall that logarithmic expressions can
be re-written as exponential expressions. We'll apply the following
relationship to each individual term in the expression.
If we have a logarithmic expression in the form
it can be rewritten as . Let's apply
this to the first term, .
tells us that raised to some number, , equals .
raised to the power of is , so is equal to . We can
repeat this process with the next term, .
tells us that raised to some number, , equals .
raised to the power of is , so is equal to . Repeat
this one more time for the last term, .
tells us that raised to some number, , equals
raised to the power of is , so is Substitute the
calculated in for the log expressions to evaluate.
Once the values are substituted, evaluate to addition.
The expression evaluates to 7.
CONCEPT
Introduction to Logarithms
4
Consider the function .
Find the formula for the inverse of this function.
log subscript b open parentheses y close parentheses
equals x
b to the power of x equals y
x
x
x
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RATIONALE
To find the inverse of a function, you can write the function
as , swap the variables and , and then
rewrite the equation with on one side. First, start by swapping
with .
Here is the function written as an equation where
. Next, we will swap the variables,
and .
Now that the variables are swapped, we will manipulate this equation to place on one side
of the equation. We'll start by squaring both sides to undo the radical.
When a square root is squared, the result is the expression under the radical. Next, we will add
to both sides to undo the subtraction of .
We have now isolated the variable to one side of the equation, which results in the inverse
function.
This is the inverse of .
CONCEPT
Finding the Inverse of a Function
5
Consider the function .
What are the domain and range of this function?
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