Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #231 : Functions And Lines

What is the common difference in this sequence?

\(\displaystyle -3,0,3,6,9,12,...\)

Possible Answers:

\(\displaystyle 5\)

\(\displaystyle 4\)

\(\displaystyle 2\)

\(\displaystyle 3\)

\(\displaystyle 1\)

Correct answer:

\(\displaystyle 3\)

Explanation:

The common difference is the distance between each number in the sequence. Notice that each number is 3 away from the previous number.

\(\displaystyle \\12-9=3\\ 9-6=3\\ 6-3=3\\3-0=3\\0-(-3)=3\)

Example Question #6 : How To Find The Common Difference In Sequences

What is the common difference in the following sequence?

\(\displaystyle 3,11,19,27,35\)

Possible Answers:

\(\displaystyle 4\)

\(\displaystyle 16\)

\(\displaystyle 19\)

\(\displaystyle 8\)

Correct answer:

\(\displaystyle 8\)

Explanation:

What is the common difference in the following sequence?

\(\displaystyle 3,11,19,27,35\)

Common differences are associated with arithematic sequences. 

A common difference is the difference between consecutive numbers in an arithematic sequence. To find it, simply subtract the first term from the second term, or the second from the third, or so on...

\(\displaystyle 11-3=8\)

\(\displaystyle 19-11=8\)

See how each time we are adding 8 to get to the next term? This means our common difference is 8.

Example Question #7 : How To Find The Common Difference In Sequences

What is the common difference in the following sequence:  \(\displaystyle [-\frac{3}{4}, -\frac{1}{4}, \frac{1}{4},...]\)

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle \frac{1}{4}\)

\(\displaystyle 2\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle \frac{1}{2}\)

Explanation:

The common difference in this set is the linear amount spaced between each number in the set.

Subtract the first number from the second number.

\(\displaystyle -\frac{1}{4}-(-\frac{3}{4}) = -\frac{1}{4}+\frac{3}{4} = \frac{2}{4} = \frac{1}{2}\)

Check this number by subtracting the second number from the third number.

\(\displaystyle \frac{1}{4}-(-\frac{1}{4}) = \frac{1}{4}+\frac{1}{4} = \frac{2}{4}=\frac{1}{2}\)

Each spacing, or common difference is:  \(\displaystyle \frac{1}{2}\)

Example Question #2 : How To Find The Common Difference In Sequences

What is the common difference?  \(\displaystyle [\frac{1}{3}, \frac{1}{4},\frac{1}{5}, \frac{1}{6}... ]\)

Possible Answers:

\(\displaystyle \frac{2}{15}\)

\(\displaystyle \frac{1}{12}\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle \textup{No common difference.}\)

\(\displaystyle \frac{1}{7}\)

Correct answer:

\(\displaystyle \textup{No common difference.}\)

Explanation:

The common difference can be determined by subtracting the first term with the second term, second term with the third term, and so forth.   The common difference must be similar between each term.

\(\displaystyle \frac{1}{3}- \frac{1}{4}= \frac{4}{12}-\frac{3}{12} = \frac{1}{12}\)

The distance between the first and second term is \(\displaystyle \frac{1}{12}\).

\(\displaystyle \frac{1}{4}-\frac{1}{5} = \frac{5}{20}-\frac{4}{20}=\frac{1}{20}\)

The distance between the second and third term is \(\displaystyle \frac{1}{20}\).

\(\displaystyle \frac{1}{5}-\frac{1}{6} = \frac{6}{30} -\frac{5}{30} = \frac{1}{30}\)

The distance between the third and fourth term is \(\displaystyle \frac{1}{30}\).

The fractions may seem as though they have a common difference since the denominators are increasing by one for each term, but there is no common difference among the numbers.

The answer is:  \(\displaystyle \textup{No common difference.}\)

Example Question #9 : How To Find The Common Difference In Sequences

What is the common difference in the following set of data?   \(\displaystyle [-8,-3,2,...]\)

Possible Answers:

\(\displaystyle -1\)

\(\displaystyle 1\)

\(\displaystyle 11\)

\(\displaystyle -5\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle 5\)

Explanation:

In order to determine the common difference, subtract the first term from the second term.

\(\displaystyle -3-(-8) = -3+8=5\)

Verify that this is the same for the difference of the third and second terms.

\(\displaystyle 2-(-3)=2+3 = 5\)

The set of data is increasing at increments of five.  

The common difference is:  \(\displaystyle 5\)

Example Question #1 : How To Find Consecutive Integers

The product of two consective positive odd integers is 143. Find both integers.

Possible Answers:

\(\displaystyle 11\ and\ 13\)

\(\displaystyle 9\ and\ 11\)

\(\displaystyle 13\ and\ 15\)

\(\displaystyle -11\ and\ -13\)

\(\displaystyle 15\ and\ 17\)

Correct answer:

\(\displaystyle 11\ and\ 13\)

Explanation:

If \(\displaystyle n\) is one odd number, then the next odd number is \(\displaystyle n+2\). If their product is 143, then the following equation is true.

\(\displaystyle n(n+2) =143\)

Distribute into the parenthesis.

\(\displaystyle n^2+2n=143\)

Subtract 143 from both sides.

\(\displaystyle n^{2} + 2n - 143 = 0\)

This can be solved by factoring, or by the quadratic equation. We will use the latter.

\(\displaystyle n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

\(\displaystyle n=\frac{-2 \pm \sqrt{4-(4(1)(-143)}}{2} = \frac{-2 \pm \sqrt{576}}{2}=\frac{-2 \pm 24}{2}\)

\(\displaystyle n=\frac{22}{2}\ or\ n=\frac{-26}{2}\)

\(\displaystyle n=11\ or\ n=-13\)

We are told that both integers are positive, so \(\displaystyle n=11\).

The other integer is \(\displaystyle n+2\).

\(\displaystyle 11+2=13\)

Example Question #5 : Arithmetic Series

Write a rule for the following arithmetic sequence:

\(\displaystyle \left \{ 2,5,8,11,14,17,20\right \}\)

Possible Answers:

\(\displaystyle x_n=2+(n-3)\)

\(\displaystyle x_n=-1+3(n+2)\)

\(\displaystyle x_n=-2+3(n-1)\)

\(\displaystyle x_n=3+2(n-1)\)

\(\displaystyle x_n=2+3(n-1)\)

Correct answer:

\(\displaystyle x_n=2+3(n-1)\)

Explanation:

Know that the general rule for an arithmetic sequence is

\(\displaystyle x_n=a+d(n-1)\),

where \(\displaystyle a\) represents the first number in the sequence, \(\displaystyle d\) is the common difference between consecutive numbers, and \(\displaystyle n\) is the \(\displaystyle n\)-th number in the sequence.  

In our problem, \(\displaystyle a=2\).

Each time we move up from one number to the next, the sequence increases by 3.  Therefore, \(\displaystyle d=3\)

The rule for this sequence is therefore \(\displaystyle x_n=2+3(n-1)\).

Example Question #2 : How To Find Consecutive Integers

If the rule of some particular sequence is written as

\(\displaystyle x_n=1+3n^2\),

find the first five terms of this sequence

Possible Answers:

\(\displaystyle 4,13,28,49,76\)

\(\displaystyle 4,17,36,49,33\)

\(\displaystyle 0,4,13,28,49\)

none of these

\(\displaystyle 4,13,7,9,21\)

Correct answer:

\(\displaystyle 4,13,28,49,76\)

Explanation:

The first term for the sequence is where \(\displaystyle n=1\). Thus,

\(\displaystyle x_1=1+3(1)^2=4\)

So the first term is 4.  Repeat the same thing for the second \(\displaystyle \left ( n=2 \right )\), third \(\displaystyle \left ( n=3 \right )\), fourth \(\displaystyle \left ( n=4 \right )\), and fifth \(\displaystyle \left ( n=5 \right )\) terms.

\(\displaystyle x_2=1+3(2)^2=13\)

\(\displaystyle x_3=28\)

\(\displaystyle x_4=49\)

\(\displaystyle x_5=76\)

We see that the first five terms in the sequence are

\(\displaystyle 4,13,28,49,76\)

Example Question #3 : How To Find Consecutive Integers

What are three consecutive numbers that are equal to \(\displaystyle 33\)?

Possible Answers:

\(\displaystyle 1, 11, 21\)

\(\displaystyle 6, 11, 16\)

\(\displaystyle 9, 11, 13\)

\(\displaystyle 10, 11, 12\)

\(\displaystyle 8, 11, 14\)

Correct answer:

\(\displaystyle 10, 11, 12\)

Explanation:

When finding consecutive numbers assign the first number a variable.

If the first number is assigned the letter n, then the second number that is consecutive must be \(\displaystyle n + 1\) and the third number must be \(\displaystyle n + 2\).

Write it out as an equation and it should look like:

\(\displaystyle n + n + 1 + n + 2 = 33\)

Simplify the equation then,

\(\displaystyle 3n + 3 = 33\)

        \(\displaystyle - 3\)     \(\displaystyle - 3\)

\(\displaystyle 3n = 30\)

\(\displaystyle n = 10\)

If \(\displaystyle n = 10\) then \(\displaystyle n + 1 = 10 + 1 = 11\)

And \(\displaystyle n + 2=10 + 2 = 12\)

So the answer is \(\displaystyle 10, 11, 12\)

Example Question #1 : How To Find Consecutive Integers

The sum of five odd consecutive numbers add to \(\displaystyle 935\). What is the fourth largest number?

Possible Answers:

\(\displaystyle 189\)

\(\displaystyle 185\)

\(\displaystyle 171\)

\(\displaystyle 191\)

\(\displaystyle 201\)

Correct answer:

\(\displaystyle 185\)

Explanation:

Let the first number be \(\displaystyle x\).

If \(\displaystyle x\) is an odd number, the next odd numbers will be:

\(\displaystyle x+2\)\(\displaystyle x+4\)\(\displaystyle x+6\), and \(\displaystyle x+8\)

The fourth highest number would then be: \(\displaystyle x+2\)

Set up an equation where the sum of all these numbers add up to \(\displaystyle 935\).

\(\displaystyle x+(x+2)+(x+4)+(x+6)+(x+8) = 935\)

Simplify this equation.

\(\displaystyle 5x +20= 935\)

Subtract 20 from both sides.

\(\displaystyle 5x +20-20= 935-20\)

Simplify both sides.

\(\displaystyle 5x= 915\)

Divide by five on both sides.

\(\displaystyle \frac{5x}{5}= \frac{915}{5}\)

\(\displaystyle x=183\)

Corresponding to the five numbers, the set of five consecutive numbers that add up to \(\displaystyle 935\) are: \(\displaystyle [183,185,187,189,191]\)

The fourth largest number would be \(\displaystyle 185\).

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