Deck 14: Additional Topics in Algebra

Question 1

Find the right part of the equality. $$4 + 5 + 6 + \cdots + ( n + 3 ) =$$

A) $\frac { n ( 8 n + 7 ) } { 3 }$
B) $\frac { n ( n + 7 ) } { 2 }$
C) $\frac { n ( 5 n + 7 ) } { 3 }$
D) $n ( 5 n - 1 )$
E) $\frac { n ( n + 7 ) } { n }$

Answer 1

$\frac { n ( n + 7 ) } { 2 }$

Question 2

Carry out the expansion. (2a + b) ⁹

A) 2a ⁹ + 2,304a ⁸ b + 4,608a ⁷ b ² + 5,376a ⁶ b ³ + 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ + 672a ³ b ⁶ + 144a ² b ⁷ + 18ab ⁸ + b ⁹
B) 512 a ⁹ + 2,304a ⁸ b - 4,608a ⁷ b ² + 5,376a ⁶ b ³ - 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ - 672a ³ b ⁶ + 144a ² b ⁷ - 18ab ⁸ + b ⁹
C) 512a ⁹ + 2,304a ⁸ b + 4,610a ⁷ b ² + 5,376a ⁶ b ³ + 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ + 672a ³ b ⁶ + 144a ² b ⁷ + 18ab ⁸ + b ⁹
D) 512a ⁹ + 2,304a ⁸ b - 4,608a ⁷ b ² + 5,376a ⁶ b ³ - 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ - 672a ³ b ⁶ + 144a ² b ⁷ - 18ab ⁸ + b ⁹
E) 512a ⁹ + 2,304a ⁸ b + 4,608a ⁷ b ² + 5,376a ⁶ b ³ + 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ + 672a ³ b ⁶ + 144a ² b ⁷ + 18ab ⁸ + b ⁹

Answer 2

512a ⁹ + 2,304a ⁸ b + 4,608a ⁷ b ² + 5,376a ⁶ b ³ + 4,032a ⁵ b ⁴ + 2,016a ⁴ b ⁵ + 672a ³ b ⁶ + 144a ² b ⁷ + 18ab ⁸ + b ⁹

Question 3

Find the indicated term. $11,20,29,38 , \ldots , a _ { 12 }$

A) 101
B) 110
C) 220
D) 55
E) 111

Answer 3

110

Question 4

Evaluate the sum. $\sum _ { n = 20 } ^ { 110 } ( 2 n - 1 )$

A) $11,639$
B) $11,689$
C) $11,789$
D) $11,839$
E) $11,739$

Question 5

Evaluate and simplify. $\left( \begin{array} { l } 4 \\0\end{array} \right)$

A) 0
B) 4
C) 14
D) 9
E) 1

Question 6

Find the indicated term. $5,2 , - 1 , - 4 , \ldots , a _ { 20 }$

A) $- 49$
B) $- 51$
C) $- 26$
D) $- 52$
E) $- 56$

Question 7

Carry out the expansion. (2 - 2x) ⁶

A) 64 - 387x - 957x ² - 1,280x ³ - 960x ⁴ - 384x ⁵ - 64x ⁶
B) 64 + 384x + 960x ² - 1,280x ³ + 960x ⁴ - 384x ⁵ + 64x ⁶
C) 64 - 384x - 960x ² - 1,280x ³ - 960x ⁴ - 384x ⁵ - 64x ⁶
D) 64 - 384x + 960x ² - 1,280x ³ + 960x ⁴ - 384x ⁵ + 64x ⁶
E) 64 - 387x + 960x ² - 1,280x ³ + 960x ⁴ - 384x ⁵ + 64x ⁶

Question 8

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$ .

A) $\frac { 20 \sqrt { 30 } } { 3 }$
B) limit does not exist
C) $\frac { \sqrt { 6 } } { 3 } + 500$
D) $\frac { \sqrt { 6 } } { 3 } - 500$
E) $\frac { \sqrt { 30 } } { 75 } - 500$

Question 9

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$

A) 24
B) $\infty$
C) 0
D) 12
E) limit does not exist

Question 10

Find the sum of the first 50 terms in an arithmetic series that has first term - 18 and 50th term 129.

A) $2,900$
B) $2,885$
C) $2,655$
D) $2,775$
E) $2,995$

Question 11

For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } > n$

A) $n \geq 25$
B) $n \geq 2$
C) $n \geq 23$
D) $n \leq 23$
E) $37 \leq n \leq 54$

Question 12

For what natural numbers is the following inequality true? $n ^ { 3 } > ( n + 10 ) ^ { 2 }$

A) $n \leq 7$
B) $n \geq 8$
C) $25 \leq n \leq 25$
D) $9 \leq n \leq 25$
E) $n \geq 7$

Question 13

The sequence is defined recursively. Compute the first five terms of the sequence. $a _ { 1 } = - 2 ; a _ { n } = \sqrt { a _ { 2 } ^ { n - 1 } + 1 } , n \geq 2$

A) $- 2 , \sqrt { 5 } , \sqrt { 6 } , \sqrt { 5 } , \sqrt { 2 }$
B) $- 2 , \sqrt { 10 } , \sqrt { 6 } , \sqrt { 7 } , 4 \sqrt { 2 }$
C) $- 2 , \sqrt { 5 } , \sqrt { 6 } , \sqrt { 7 } , 2 \sqrt { 2 }$
D) $- 2 , \sqrt { 5 } , \sqrt { 3 } , \sqrt { 7 } , 2 \sqrt { 8 }$
E) $- 2 , \sqrt { 5 } , \sqrt { 3 } , \sqrt { 7 } , \sqrt { 2 }$

Question 14

Find the coefficient of the term containing a ⁴ in the expansion of $( \sqrt { a } - \sqrt { x } ) ^ { 16 }$ .

A) 12,870
B) 495
C) 12,871
D) 25,740
E) 1,839

Question 15

Find the right part of the equality. $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) =$ ________

A) $\frac { n ^ { 2 } ( n + 3 ) } { 4 }$
B) $\frac { n ( n + 1 ) ( n + 2 ) } { 3 }$
C) $\frac { n \left( 4 n ^ { 2 } + 6 n - 1 \right) } { 3 }$
D) $\frac { n ^ { 2 } ( n - 1 ) ( n + 3 ) } { 2 }$

Question 16

Find the third term in the expansion of (a - b) ²⁷ .

A) - 17,550a ⁴ b ²³
B) 17,550a ²³ b ⁴
C) - 17,550a ²³ b ⁴
D) 17,551a ²³ b ⁴
E) - 17,551a ²³ b ⁴

Question 17

The sum of three consecutive terms in an arithmetic sequence is 30, and their product is 750. Find the three terms. (Suggestion: Let x denote the middle term and d the common difference.)

A) 5, 9, 13
B) 4, 9, 14
C) 5, 10, 15
D) 1, 10, 15
E) 6, 10, 14

Question 18

Express as a fraction. $0 . \overline { 71 }$

A) $\frac { 71 } { 99 }$
B) $\frac { 71 } { 100 }$
C) $\frac { 71 } { 110 }$
D) $\frac { 71 } { 183 }$
E) $\frac { 71 } { 97 }$

Question 19

Find the common difference d for the following arithmetic sequence. 12, 7, 2, - 3, ...

A) 5
B) - 5
C) 3
D) - 3
E) - 6

Question 20

Find the right part of the equality. $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } =$ ________

A) $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$
B) $\frac { n ( n + 1 ) ( n + 3 ) } { 4 }$
C) $\frac { n ( n + 1 ) } { 2 }$
D) $\frac { ( n + 1 ) ( n + 2 ) } { 3 }$
E) none of the above

Question 21

Use the position function $s ( t ) = - 16 t ^ { 2 } + 120$ to find the velocity in feet/second at time $t = 2$ seconds. The velocity at time $t = c$ seconds is given by $\lim _ { t \rightarrow c } \frac { [ s ( c ) - s ( t ) ] } { ( c - t ) }$

A) limit does not exist
B) 32 feet/second
C) 64 feet/second
D) 0 feet/second
E) -64 feet/second

Question 22

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 13 + y } - \sqrt { 13 } } { y }$ .

A) $\frac { \sqrt { 13 } } { 13 }$
B) limit does not exist
C) $\frac { \sqrt { 13 } } { 2 }$
D) $\frac { \sqrt { 13 } } { 26 }$
E) 0

Question 23

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 5 } } { x - 2 }$ by direct substitution.

A) $\frac { \sqrt { 11 } } { 4 }$
B) $\frac { \sqrt { 11 } } { 2 }$
C) $\frac { \sqrt { 6 } } { 4 }$
D) $\sqrt { 11 }$
E) $\frac { 1 } { 4 }$

Question 24

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists).

A) 0
B) 3
C) -3
D) limit does not exist
E) 6

Question 25

Find $\lim _ { t \rightarrow 5 } \frac { t ^ { 3 } - 125 } { t - 5 }$ .

A) limit does not exist
B) 5
C) 50
D) 75
E) 15