High School Math : High School Math

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #11 : Sequences And Series

Find the sum of all even integers from  to .

Possible Answers:

Correct answer:

Explanation:

The formula for the sum of an arithmetic series is

,

where  is the number of terms in the series,  is the first term, and  is the last term.

We know that there are  terms in the series. The first term is  and the last term is . Our formula becomes:

Example Question #1 : Finding Partial Sums In A Series

Find the sum of all even integers from  to .

Possible Answers:

Correct answer:

Explanation:

The formula for the sum of an arithmetic series is

,

where  is the number of terms in the series,  is the first term, and  is the last term.

 

Example Question #2063 : High School Math

Find the sum of the even integers from  to .

Possible Answers:

Correct answer:

Explanation:

The sum of even integers represents an arithmetic series.

The formula for the partial sum of an arithmetic series is

,

where  is the first value in the series,  is the number of terms, and  is the difference between sequential terms.

Plugging in our values, we get:

Example Question #31 : Sequences And Series

Find the value for 

Possible Answers:

Correct answer:

Explanation:

To best understand, let's write out the series. So

We can see this is an infinite geometric series with each successive term being multiplied by .

A definition you may wish to remember is

 where  stands for the common ratio between the numbers, which in this case is  or . So we get

 

Example Question #2 : Finding Sums Of Infinite Series

Evaluate:

Possible Answers:

The series does not converge.

Correct answer:

Explanation:

This is a geometric series whose first term is   and whose common ratio is . The sum of this series is:

Example Question #2 : Sums Of Infinite Series

Evaluate:

Possible Answers:

The series does not converge.

Correct answer:

Explanation:

This is a geometric series whose first term is   and whose common ratio is . The sum of this series is:

Example Question #1 : Limits

Possible Answers:

Correct answer:

Explanation:

A limit describes what -value a function approaches as  approaches a certain value (in this case, ). The easiest way to find what -value a function approaches is to substitute the -value into the equation.

Substituting  for  gives us an undefined value (which is NOT the same thing as 0). This means the function is not defined at that point. However, just because a function is undefined at a point doesn't mean it doesn't have a limit. The limit is simply whichever value the function is getting close to.

One method of finding the limit is to try and simplify the equation as much as possible:

As you can see, there are common factors between the numerator and the denominator that can be canceled out. (Remember, when you cancel out a factor from a rational equation, it means that the function has a hole -- an undefined point -- where that factor equals zero.)

After canceling out the common factors, we're left with:

Even though the domain of the original function is restricted ( cannot equal ), we can still substitute into this simplified equation to find the limit at

Example Question #1 : Finding One Sided Limits

Let .

Find .

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

1overx

This is a graph of . We know that  is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as  approaches 0 from the left,  approaches negative infinity. 

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful. 

 is actually infinity, not negative infinity. 

Example Question #3 : Limits

Evaluate the limit below:

 

Possible Answers:

1

0

Correct answer:

Explanation:

 will approach when approaches , so  will be of type  as shown below:

 

 

So, we can apply the L’ Hospital's Rule:

 

 

since:

hence:

Example Question #51 : Pre Calculus

Possible Answers:

Correct answer:

Explanation:

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