Sequences

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SAT Subject Test in Math I › Sequences

1 - 10

What is the next number in the sequence?

CORRECT

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

An arithmetic sequence begins as follows:

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

Give the tenth term of this sequence.

$\frac{3}{5}$

CORRECT

$\frac{7}{10}$

$\frac{2}{3}$

$\frac{19}{30}$

$\frac{11}{15}$

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 :

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

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

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.

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

$-12 i \sqrt{2}$

CORRECT

$-12 \sqrt{2}$

$-18 \sqrt{2}$

$-18 i \sqrt{2}$

None of these

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.

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

Insufficient information is given to answer the question.

CORRECT

$265\frac{1}{2}$

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

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

$3r^{2} =108$

$r^{2} = 36$

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.

A geometric sequence begins as follows:

Give the next term of the sequence.

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

CORRECT

$\frac{1}{4} i$

$\frac{1}{4}$

$-\frac{1}{4}$

None of the other choices gives the correct response.

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:

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

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?

$2 \sqrt{15}$

CORRECT

$6 \sqrt{5}$

$4 \sqrt{10}$

None of these

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,

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

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, $a_{2} = 2 \sqrt{15}$ .

A geometric sequence begins as follows:

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

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

$18 \sqrt{5}$

CORRECT

$10 \sqrt{5}$

$17 \sqrt{5}$

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:

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

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

$-\frac{3}{2}$

CORRECT

$\frac{3}{2}$

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.

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?