Sequences - SAT Subject Test in Math I

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Question

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?

Answer

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:

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Question

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

Answer

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

$3r^{2} =108$

$r^{2} = 36$

or .

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

If , 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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Question

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

Answer

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

$2r^{2} = 50$

$r^{2} = 25$

or .

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

If , 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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Question

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

Answer

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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Question

What is the next number in the sequence?

Answer

The first number is multiplied by three

$\left ( 23=6 \right )$ .

Then it is divide by two

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

The following is multiplied by three

$\left ( 33=9\right )$

then divided by two

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

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

.

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Question

An arithmetic sequence begins as follows:

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

Give the tenth term of this sequence.

Answer

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 :

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

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

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 $a_{1}= \frac{3}{10}$ , , 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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Question

A geometric sequence begins as follows:

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

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

Answer

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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Question

A geometric sequence begins as follows:

Give the next term of the sequence.

Answer

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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Question

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

Answer

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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Question

The first and third terms of a geometric sequence comprising only positive elements are $\sqrt{20}$ and $\sqrt{180}$ , respectively. In simplest form, which of the following is its second term?

Answer

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,

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

Setting $a_{1}= \sqrt{20}, a_{3} = \sqrt{180}$ , and applying the Quotient of Radicals Rule:

$r ^{2} = \frac{a_{3}}{a_{1}} = \frac{\sqrt{180}}{\sqrt{20}} = \sqrt{9} = 3$

Taking the square root of both sides:

$r = \pm \sqrt{3}$

Substituting, and applying the Product of Radicals Rule:

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

$a_{2} = \pm \sqrt{3} \cdot \sqrt{20 } =\pm \sqrt{60 } = \pm \sqrt{4} \cdot \sqrt{15}= \pm 2 \sqrt{15}$

Since all elements of the sequence are positive, $a_{2} = 2 \sqrt{15}$ .

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Question

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

Answer

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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