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SAT II Math I : Sequences

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #1 : Sequences

The first two numbers of a sequence are, in order, 1 and 4. Each successive element is formed by adding the previous two. What is the sum of the first six elements of the sequence?

Possible Answers:

Correct answer:

Explanation:

The first six elements are as follows:

$a_{1} = 1 \;$

$a_{2}= 4$

$a_{3} = 1 + 4 = 5$

$a_{4} = 4 + 5 = 9$

$a_{5} = 5+ 9 = 14$

$a_{6} = 9 + 14 = 23$

Add them:

1 + 4 + 5 + 9 + 14 + 23 = 56

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Example Question #1 : Sequences

The first and third terms of a geometric sequence are 3 and 108, respectively. All What is the sixth term?

Possible Answers:

Insufficient information is given to answer the question.

$-265\frac{1}{2}$

-23,328

23,328

$265\frac{1}{2}$

Correct answer:

Insufficient information is given to answer the question.

Explanation:

Let the common ratio of the sequence be . Then The first three terms of the sequence are $3,3r,3r^{2}$ . The third term is 108, so

$3r^{2} =108$

$r^{2} = 36$

r = 6 or r = -6 .

The common ratio can be either - not enough information exists for us to determine which.

The sixth term is

$a_{1}r^{6-1} = 3r^{5}$

If r = 6 , the seventh term is $3 \cdot 6^{5}= 3 \cdot 7,776 = 23,328$ .

If r =- 6 , the seventh term is $3 \cdot \left (-6 \right )^{5}= 3 \cdot \left (-7,776 \right )= -23,328$ .

Therefore, not enough information exists to determine the sixth term of the sequence.

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Example Question #1 : Sequences

The first and third terms of a geometric sequence are 2 and 50, respectively. What is the seventh term?

Possible Answers:

-31,250

Insufficient information is given to answer the question.

146

31,250

Correct answer:

31,250

Explanation:

Let the common ratio of the sequence be . Then The first three terms of the sequence are $2,2r,2r^{2}$ . The third term is 50, so

$2r^{2} = 50$

$r^{2} = 25$

r = 5 or r = -5 .

Not enough information is given to choose which one is the common ratio. But the seventh term is

$a_{1}r^{7-1} = 2r^{6}$

If r = 5 , the seventh term is $2 \cdot 5^{6}= 2 \cdot 15,625 = 31,250$ .

If r = - 5 , the seventh term is $2 \cdot\left ( -5 \right )^{6}= 2 \cdot 15,625 = 31,250$ .

Either way, the seventh term is 31,250.

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Example Question #2 : Sequences

The sum of 3 odd consecutive numbers is 345. What is the largest number in the sequence?

Possible Answers:

121

103

117

119

Correct answer:

117

Explanation:

When you are dealing with arithmetic means, it is best to define one number in the sequence as x and every other number relative to x.

Because we are trying to find the largest of three numbers, let's define x as the largest number in the equation. Because each number is a consecutive odd number, we must subtract 2 to get to the next number in the sequence.

x: largest number in sequence

x-2: middle number in sequence

x-4: smallest number in sequence

Now, let's make an equation finds the sum of all the numbers in the sequence and set it equal to 354.

$\small x+(x-2)+(x-4)=345$

$\small 3x-6+6=345+6$

$\small \frac{3x}{3}=\frac{351}{3}$

$\small x=117$

117 is the largest number in the sequence.

To check yourself, you can add up the numbers in the sequence {113, 115, 117}.

$\small 113+115+117=345$

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Example Question #5 : Sequences

2, 6, 3, 9, 4.5, ?

What is the next number in the sequence?

Possible Answers:

2.25

13.5

Correct answer:

13.5

Explanation:

The first number is multiplied by three

$\left ( 2*3=6 \right )$ .

Then it is divide by two

$\left (6/2=3 \right )$ .

The following is multiplied by three

$\left ( 3*3=9\right )$

then divided by two

$\left ( 9/2=4.5\right )$ .

That makes the next step to multiply by three which gives us

4.5*3=13.5 .

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Example Question #62 : Mathematical Relationships

An arithmetic sequence begins as follows:

$0.3, \frac{1}{3} ,...$

Give the tenth term of this sequence.

Possible Answers:

$\frac{3}{5}$

$\frac{11}{15}$

$\frac{7}{10}$

$\frac{2}{3}$

$\frac{19}{30}$

Correct answer:

$\frac{3}{5}$

Explanation:

Rewrite the first term in fraction form: $0.3 = \frac{3}{10}$ .

The sequence now begins

$\frac{3}{10}, \frac{1}{3}$ ,...

Rewrite the terms with their least common denominator, which is LCM (3, 10) = 30 :

$\frac{3}{10} = \frac{9}{30}$

$\frac{1}{3} = \frac{10}{30}$

The common difference of the sequence is the difference of the second and first terms, which is

$d = \frac{10}{30} - \frac{9}{30} = \frac{1}{30}$ .

The rule for term $a_{n}$ of an arithmetic sequence, given first term $a _{1}$ and common difference , is

$a_{n} = a_{1} + (n-1)d$ ;

Setting $a_{1}= \frac{3}{10}$ , n = 10 , and $d = \frac{1}{30}$ , we can find the tenth term $a_{10}$ by evaluating the expression:

$a_{10} = \frac{3}{10} + (10-1) \frac{1}{30 }$

$= \frac{3}{10} + 9 \cdot \frac{1}{30 }$

$= \frac{3}{10} + \frac{9}{30 }$

$= \frac{9}{30} + \frac{9}{30 }$

$= \frac{18}{30 }$

$= \frac{3}{5}$ ,

the correct response.

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Example Question #6 : Sequences

A geometric sequence begins as follows:

$\sqrt{20} , \sqrt{180},...$

Express the next term of the sequence in simplest radical form.

Possible Answers:

$17 \sqrt{5}$

$10 \sqrt{5}$

$18 \sqrt{5}$

Correct answer:

$18 \sqrt{5}$

Explanation:

Using the Product of Radicals principle, we can simplify the first two terms of the sequence as follows:

$a_{1}=\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}$

$a_{2}= \sqrt{180} = \sqrt{36} \cdot \sqrt{5} = 6 \sqrt{5}$

The common ratio of a geometric sequence is the quotient of the second term and the first:

$r = \frac{a_{2}}{a_{1}}= \frac{6 \sqrt{5}}{2\sqrt{5} } = 3$

Multiply the second term by the common ratio to obtain the third term:

$a_{3} = r a_{2} = 3 \cdot 6 \sqrt{5} = 18 \sqrt{5}$

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Example Question #64 : Mathematical Relationships

The second and third terms of a geometric sequence are 3 i and , respectively. Give the first term.

Possible Answers:

$-\frac{3}{2}$

$\frac{3}{2}$

Correct answer:

$-\frac{3}{2}$

Explanation:

The common ratio of a geometric sequence is the quotient of the third term and the second:

$r =\frac{a_{3} }{a_{2}} = \frac{-6}{3i} = \frac{-2}{i}$

Multiplying numerator and denominator by , this becomes

$r = \frac{-2}{i} = \frac{-2(-i)}{i (-i)} = \frac{-2i}{ -i^{2}} = \frac{-2i}{ -(-1 )} = \frac{-2i}{1} = -2i$

The second term of the sequence is equal to the first term multiplied by the common ratio:

$r \cdot a_{1} = a_{2}$ .

so equivalently:

$a_{1} =\frac{ a_{2}}{r}$

Substituting:

$a_{1} =\frac{ 3i}{-2i} = -\frac{3}{2}$ ,

the correct response.

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Example Question #1 : Sequences

A geometric sequence begins as follows:

4 i , 1 ,...

Give the next term of the sequence.

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

None of the other choices gives the correct response.

$-\frac{1}{4} i$

$\frac{1}{4} i$

Correct answer:

$-\frac{1}{4} i$

Explanation:

The common ratio of a geometric sequence is the quotient of the second term and the first:

$r = \frac{a_{2}}{a_{1}}= \frac{1}{4i}$

Simplify this common ratio by multiplying both numerator and denominator by :

$r = \frac{1}{4i} = \frac{1 \cdot i}{4i \cdot i } = \frac{i}{4 i^{2}} = \frac{i}{4 (-1)} = \frac{i}{-4 } = -\frac{1}{4} i$

Multiply the second term by the common ratio to obtain the third term:

$a_{3} = r a_{2} = -\frac{1}{4} i \cdot 1 = -\frac{1}{4} i$

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Example Question #66 : Mathematical Relationships

A geometric sequence has as its first and third terms and 24, respectively. Which of the following could be its second term?

Possible Answers:

None of these

$-18 \sqrt{2}$

$-18 i \sqrt{2}$

$-12 \sqrt{2}$

$-12 i \sqrt{2}$

Correct answer:

$-12 i \sqrt{2}$

Explanation:

Let be the common ratio of the geometric sequence. Then

$a_{2} = r a_{1}$

and

$a_{3} = r a_{2}= r \cdot r a_{1} = r ^{2}a_{1}$

Therefore,

$r ^{2} = \frac{a_{3}}{a_{1}}$ ,

and

$r = \pm \sqrt{ \frac{a_{3}}{a_{1}}}$

Setting $a_{1}= -12, a_{3} = 24$ :

$r = \pm \sqrt{ \frac{24}{-12} } = \pm\sqrt{-2} = \pm \sqrt{-1} \cdot \sqrt{2} = \pm i \sqrt{2}$ .

Substituting for and $a_{1}$ , either

$a_{2} = r a_{1} = \pm i \sqrt{2} \cdot (-12) = \mp 12 i \sqrt{2}$ .

The second term can be either $-12 i \sqrt{2}$ or $12 i \sqrt{2}$ , the former of which is a choice.

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SAT II Math I : Sequences

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Sequences

Example Question #1 : Sequences

Example Question #1 : Sequences

Example Question #2 : Sequences

The sum of 3 odd consecutive numbers is 345. What is the largest number in the sequence?

117 is the largest number in the sequence.

Example Question #5 : Sequences

Example Question #62 : Mathematical Relationships

Example Question #6 : Sequences

Example Question #64 : Mathematical Relationships

Example Question #1 : Sequences

Example Question #66 : Mathematical Relationships

All SAT II Math I Resources

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