Deck 54: Geometric Sequences and Series

Question 1

Write an expression for the nth term of the geometric sequence.Then find the indicated term.(Round your answer to three decimal places. )? $$a _ { 1 } = 2500 , r = 1.005 , n = 60$$ ?

A) $a _ { n } = ( 1.005 ) ^ { n - 1 }$ ,About 3355.349
B) $a _ { n } = 2500 ( 1.005 ) ^ { n - 1 }$ ,About 3355.349
C) $a _ { n } = 2500 ( 1.005 ) ^ { n - 1 }$ ,About 5520.099
D) $a _ { n } = ( 1.005 ) ^ { n + 1 }$ ,About 3355.349
E) $a _ { n } = 2500 ( 1.005 ) ^ { n + 1 }$ ,About 3355.349

Answer 1

$a _ { n } = 2500 ( 1.005 ) ^ { n - 1 }$ ,About 3355.349

Question 2

Find the indicated nth term of the geometric sequence. ?
10th term : $$7,28,112 , \ldots$$ ?

A) $a _ { 10 } = 114,688$
B) $a _ { 10 } = 7,340,032$
C) $a _ { 10 } = 458,752$
D) $a _ { 10 } = 28,672$
E) $a _ { 10 } = 1,835,008$

Answer 2

$a _ { 10 } = 1,835,008$

Question 3

Determine whether the sequence is geometric.If so,find the common ratio.? $$7,1.4,0.28,0.056 , \ldots$$ ?

A)Geometric sequence, $r = 0.5$
B)Geometric sequence, $r = 0.2$
C)Geometric sequence, $r = 0.4$
D)Not a geometric sequence
E)Geometric sequence, $r = 0.3$

Answer 3

Geometric sequence, $r = 0.2$

Question 4

Find the indicated nth term of the geometric sequence. ?
22th term : $$8,32,128 , \ldots$$ ?

A) $a _ { 22 } = 35,184,372,088,832$
B) $a _ { 22 } = 549,755,813,888$
C) $a _ { 22 } = 2,199,023,255,552$
D) $a _ { 22 } = 8,796,093,022,208$
E) $a _ { 22 } = 140,737,488,355,328$

Question 5

Determine whether the sequence is geometric.If so,find the common ratio.? $$7,14,22,70 , \ldots$$ ?

A)Geometric sequence, $$r = 1$$
B)Geometric sequence, $$r = 2$$
C)Geometric sequence, $r = 3$
D)Geometric sequence, $r = 4$
E)Not a geometric sequence

Question 6

Find the first five terms of the geometric sequence.? $$a _ { 1 } = 5 , r = 3$$ ?

A) $$5,15,45,135,405$$
B) $$9,27,81,243,729$$
C) $15,45,135,405,1215$
D) $1,3,9,27,81$
E) $3,9,27,81,243$

Question 7

Find the first five terms of the geometric sequence.? $$a _ { 1 } = 6 , r = 2$$ ?

A) $4,8,16,32,64$
B) $5,10,20,40,80$
C) $12,24,48,96,192$
D) $6,12,24,48,96$
E) $10,20,40,80,160$

Question 8

Write an expression for the nth term of the geometric sequence.Then find the indicated term.(Round your answer to three decimal places. )? $$a _ { 1 } = 900 , r = 1.02 , n = 40$$ ?

A) $a _ { n } = 900 ( 1.02 ) ^ { n + 1 }$ ,About 1948.270
B) $a _ { n } = ( 1.02 ) ^ { n - 1 }$ ,About 1948.270
C) $a _ { n } = 900 ( 1.02 ) ^ { n - 1 }$ ,About 1948.270
D) $a _ { n } = 900 ( 1.02 ) ^ { n - 1 }$ ,About 1987.236
E) $a _ { n } = ( 1.02 ) ^ { n + 1 }$ ,About 1948.270

Question 9

Determine whether the sequence is geometric.If so,find the common ratio.? $$7,21,63,189 , \ldots$$ ?

A)Geometric sequence, $r = 3$
B)Not a geometric sequence
C)Geometric sequence, $r = 8$
D)Geometric sequence, $r = 7$
E)Geometric sequence, $r = 4$

Question 10

Find the sum of the finite geometric sequence.? $$\sum _ { n = 1 } ^ { 7 } 2 ^ { n - 1 }$$ ?

A)43
B)22
C)127
D)63
E)31

Question 11

Write an expression for the nth term of the geometric sequence.Then find the indicated term.(Round your answer to three decimal places. )? $$a _ { 1 } = 7 , r = \sqrt { 3 } , n = 8$$ ?

A) $a _ { n } = 7 ( \sqrt { 3 } ) ^ { n - 1 }$ ,About 15.456
B) $a _ { n } = ( \sqrt { 3 } ) ^ { n - 1 }$ ,About 982.073
C) $a _ { n } = 7 ( \sqrt { 3 } ) ^ { n + 1 }$ ,About 327.358
D) $a _ { n } = 7 ( \sqrt { 3 } ) ^ { n - 1 }$ ,About 327.358
E) $a _ { n } = ( \sqrt { 3 } ) ^ { n + 1 }$ ,About 15.456

Question 12

Write an expression for the nth term of the geometric sequence.Then find the indicated term.? $$a _ { 1 } = 4 , r = \frac { 1 } { 6 } , n = 10$$ ?

A) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ , $\frac { 1 } { 2,519,424 }$
B) $a _ { n } = 4 \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$ , $\frac { 1 } { 2,519,424 }$
C) $a _ { n } = 4 \left( \frac { 1 } { 6 } \right) ^ { n + 1 }$ , $\frac { 1 } { 2,519,424 }$
D) $a _ { n } = 6 \left( \frac { 1 } { 2 } \right) ^ { n + 1 }$ , $\frac { 1 } { 2,519,424 }$
E) $a _ { n } = \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$ , $\frac { 1 } { 2,519,424 }$

Question 13

Find the first five terms of the geometric sequence.? $$a _ { 1 } = 4 , r = 2 x$$ ?

A) $4,2 x , 2 x ^ { 2 } , 2 x ^ { 3 } , 2 x ^ { 4 }$
B) $4,8 x , 16 x ^ { 2 } , 32 x ^ { 3 } , 64 x ^ { 4 }$
C) $4,2 x ^ { 2 } , 2 x ^ { 3 } , 2 x ^ { 4 } , 2 x ^ { 5 }$
D) $4,8 x ^ { 2 } , 16 x ^ { 3 } , 32 x ^ { 4 } , 64 x ^ { 5 }$
E) $4 , x ^ { 2 } , x ^ { 3 } , x ^ { 4 } , x ^ { 5 }$

Question 14

Find the first five terms of the geometric sequence.? $$a _ { 1 } = 4 , r = e$$ ?

A) $4 , e , 2 e , 3 e , 4 e$
B) $4 , e ^ { 2 } , e ^ { 3 } , e ^ { 4 } , e ^ { 5 }$
C) $e , e , e , e , e$
D) $4 , e , 2 e ^ { 2 } , 3 e ^ { 3 } , 4 e ^ { 4 }$
E) $4,4 e , 4 e ^ { 2 } , 4 e ^ { 3 } , 4 e ^ { 4 }$

Question 15

Find the indicated nth term of the geometric sequence. ?
9th term : $$13,39,117 , \ldots$$ ?

A) $a _ { 9 } = 3,159$
B) $a _ { 9 } = 255,879$
C) $a _ { 9 } = 28,431$
D) $a _ { 9 } = 85,293$
E) $a _ { 9 } = 9,477$

Question 16

Find the first five terms of the geometric sequence.? $$a _ { 1 } = 4 , r = \pi$$ ?

A) $$4,4 \pi , 4 \pi ^ { 2 } , 4 \pi ^ { 3 } , 4 \pi ^ { 4 }$$
B) $4,4 \pi ^ { 2 } , 4 \pi ^ { 3 } , 4 \pi ^ { 4 } , 4 \pi ^ { 5 }$
C) $4 , \pi ^ { 2 } , \pi ^ { 3 } , \pi ^ { 4 } , \pi ^ { 5 }$
D) $4 , \pi , \pi ^ { 2 } , \pi ^ { 3 } , \pi ^ { 4 }$
E) $4 , \pi , 4 \pi ^ { 2 } , 4 \pi ^ { 3 } , 4 \pi ^ { 4 }$

Question 17

Find the indicated nth term of the geometric sequence. ?
7th term : $$3,21,147 , \ldots$$ ?

A) $a _ { 7 } = 50,421$
B) $a _ { 7 } = 7,203$
C) $a _ { 7 } = 17,294,403$
D) $a _ { 7 } = 352,947$
E) $a _ { 7 } = 2,470,629$

Question 18

Find the sum of the finite geometric sequence? $$\sum _ { n = 1 } ^ { 6 } ( - 6 ) ^ { n - 1 }$$ ?

A)-6,665
B)-9,331
C)-185
D)1,555
E)1,111

Question 19

Determine whether the sequence is geometric.If so,find the common ratio.? $$4,20,100,500 , \ldots$$ ?

A)Geometric sequence, $r = 4$
B)Geometric sequence, $r = 6$
C)Geometric sequence, $r = 5$
D)Not a geometric sequence
E)Geometric sequence, $r = 7$

Question 20

Determine whether the sequence is geometric.If so,find the common ratio.? $$55,50,45,40 , \ldots$$ ?

A)Geometric sequence, $$r = 4$$
B)Geometric sequence, $r = 6$
C)Geometric sequence, $$r = 5$$
D)Geometric sequence, $r = 7$
E)Not a geometric sequence

Question 21

Find the sum of the infinite geometric series.? $$\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 9 } { 10 } \right) ^ { n }$$ ?

A)91
B)92
C)88
D)89
E)90

Question 22

Use summation notation to write the sum.? $$8 + 16 + 32 + \ldots + 1024$$ ?

A) $\sum _ { n = 1 } ^ { 8 } 8 ( 2 ) ^ { n + 1 }$
B) $\sum _ { n = 1 } ^ { 8 } 8 ( 2 ) ^ { n - 1 }$
C) $\sum _ { n = 1 } ^ { 8 } 8 ( 2 ) ^ { n }$
D) $$\sum _ { n = 1 } ^ { 7 } 8 ( 2 ) ^ { n - 1 }$$
E) $\sum _ { n = 1 } ^ { 8 } ( 2 ) ^ { n - 1 }$

Question 23

Find the sum of the infinite geometric series.? $$\sum _ { n = 0 } ^ { \infty } \left( - \frac { 1 } { 8 } \right) ^ { n }$$ ?

A) $\frac { 8 } { 9 }$
B) $- \frac { 8 } { 9 }$
C) $- \frac { 9 } { 8 }$
D) $\frac { 9 } { 10 }$
E) $\frac { 9 } { 8 }$

Question 24

A principal of $5000 is invested at 4% interest.Find the amount after 10 years if the interest is compounded semi-annually. ​

A)$7431.74
B)$7433.74
C)$7432.74
D)$7430.74
E)$7429.74

Question 25

A principal of $2000 is invested at 5% interest.Find the amount after 10 years if the interest is compounded annually. ​

A)$3261.79
B)$3258.79
C)$3259.79
D)$3257.79
E)$3260.79

Question 26

Find the sum of the finite geometric sequence.(Round your answer to three decimal places. )? $$\sum _ { n = 0 } ^ { 6 } 900 ( 1.04 ) ^ { n }$$ ?

A)999.405
B)5,969.678
C)7,108.465
D)4,874.690
E)1,021.735

Question 27

A principal of $3000 is invested at 5% interest.Find the amount after 20 years if the interest is compounded monthly. ​

A)$8137.92
B)$8135.92
C)$8139.92
D)$8136.92
E)$8138.92

Question 28

Determine whether the sequence is geometric.If so,find the common ratio. -2,2,6,10,...

A) $\frac { 1 } { 4 }$
B)-2
C)4
D)not geometric
E)-4

Question 29

Write the first five terms of the geometric sequence. $$a _ { 1 } = - 5 , r = \frac { 1 } { 8 }$$

A) $- 5 , - \frac { 5 } { 8 } , - \frac { 5 } { 64 } , - \frac { 5 } { 512 } , - \frac { 5 } { 4,096 }$
B) $1 , - 5 , - \frac { 5 } { 8 } , - \frac { 5 } { 64 } , - \frac { 5 } { 512 }$
C)-5,-40,-320,-2,560,-20,480
D)-5,3,11,19,27
E) $- \frac { 5 } { 8 } , - \frac { 5 } { 64 } , - \frac { 5 } { 512 } , - \frac { 5 } { 4,096 } , - \frac { 5 } { 32,768 }$ ?

Question 30

A tool and die company buys a machine for $175,000 and it depreciates at a rate of 50% per year.(In other words,at the end of each year the depreciated value is 50% of what it was at the beginning of the year. )Find the depreciated value of the machine after 5 full years. ​

A)$5470.75
B)$5467.75
C)$5468.75
D)$5471.75
E)$5469.75

Question 31

Find the sum of the finite geometric sequence.(Round your answer to three decimal places. )? $$\sum _ { n = 0 } ^ { 5 } 300 ( 1.06 ) ^ { n }$$ ?

A)2,092.596
B)1,691.128
C)340.518
D)352.212
E)1,312.385

Question 32

Find the sum of the infinite geometric series.? $$\sum _ { n = 0 } ^ { \infty } 4 ( 0.8 ) ^ { n }$$ ?

A)24
B)20
C)22
D)21
E)19

Question 33

Find the sum of the infinite geometric series.(Round your answer to the nearest whole number. )? $$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n }$$ ?

A) $\frac { 8 } { 9 }$
B) $\frac { 7 } { 8 }$
C) $\frac { 9 } { 7 }$
D) $\frac { 9 } { 8 }$
E) $\frac { 8 } { 7 }$

Question 34

Determine whether the sequence is geometric.If so,find the common ratio. -3,-6,-12,-24,...

A)-3
B)-2
C) $\frac { 1 } { 2 }$
D)2
E)not geometric

Question 35

Write an expression for the nth term of the geometric sequence.Then find the indicated term.? $$a _ { 1 } = 6 , r = - \frac { 1 } { 5 } , n = 12$$ ?

A) $a _ { n } = 6 \left( - \frac { 1 } { 5 } \right) ^ { n - 1 }$ , $\frac { 6 } { 48,828,125 }$
B) $a _ { n } = 5 \left( - \frac { 1 } { 2 } \right) ^ { n + 1 }$ , $\frac { 6 } { 48,828,125 }$
C) $a _ { n } = 6 \left( - \frac { 1 } { 5 } \right) ^ { n + 1 }$ , $\frac { 6 } { 48,828,125 }$
D) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ , $\frac { 6 } { 48,828,125 }$
E) $a _ { n } = 5 \left( - \frac { 1 } { 4 } \right) ^ { n - 1 }$ , $\frac { 6 } { 48,828,125 }$

Question 36

Use summation notation to write the sum.? $$12 + 36 + 108 + \ldots + 8,748$$ ?

A) $$\sum _ { n = 1 } ^ { 7 } 12 ( 3 ) ^ { n - 1 }$$
B) $$\sum _ { n = 1 } ^ { 7 } 12 ( 3 ) ^ { n + 1 }$$
C) $\sum _ { n = 1 } ^ { 6 } 12 ( 3 ) ^ { n - 1 }$
D) $\sum _ { n = 1 } ^ { 7 } 3 ^ { n - 1 }$
E) $\sum _ { n = 1 } ^ { 7 } 12 ( 3 ) ^ { n }$

Question 37

Use summation notation to write the sum.? $$0.9 + 3.6 + 14.4 + \ldots + 921.6$$ ?

A) $\sum _ { n = 1 } ^ { 8 } 0.9 ( 4 ) ^ { n + 1 }$
B) $\sum _ { n = 1 } ^ { 6 } ( 4 ) ^ { n - 1 }$
C) $\sum _ { n = 1 } ^ { 7 } 0.9 ( 4 ) ^ { n - 1 }$
D) $\sum _ { n = 1 } ^ { 8 } 0.9 ( 4 ) ^ { n }$
E) $\sum _ { n = 1 } ^ { 6 } 0.9 ( 4 ) ^ { n - 1 }$

Question 38

A ball is dropped from a height of 15 feet.Each time it drops h feet,it rebounds 0.85h feet. Find the total vertical distance traveled by the ball.(Round your answer to two decimal places. )
​

A)187.00 ft
B)185.00 ft
C)188.00 ft
D)186.00 ft
E)184.00 ft

Question 39

A principal of $4000 is invested at 8% interest.Find the amount after 10 years if the interest is compounded quarterly. ​

A)$8835.16
B)$8833.16
C)$8836.16
D)$8834.16
E)$8832.16

Question 40

A principal of $2500 is invested at 2% interest.Find the amount after 20 years if the interest is compounded quarterly. ​

A)$3724.85
B)$3727.85
C)$3723.85
D)$3726.85
E)$3725.85

Question 41

Find the rational number representation of the repeating decimal. $$0.\overline{725}$$

A) $\frac { 725 } { 99 }$
B) $\frac { 999 } { 725 }$
C) $\frac { 725 } { 9 }$
D) $\frac { 725 } { 999 }$
E) $\frac { 725 } { 9999 }$

Question 42

Find the sum of the infinite geometric series. $$\sum _ { n = 0 } ^ { \infty } - 3 \left( - \frac { 1 } { 5 } \right) ^ { n }$$

A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $- \frac { 5 } { 2 }$
D) $- \frac { 1 } { 2 }$
E)undefined

Question 43

Find the sum of the infinite geometric series.? $$\sum _ { n = 0 } ^ { \infty } 7 \left( \frac { 1 } { 3 } \right) ^ { n }$$ ?

A)? $\frac { 2 } { 21 }$
B)? $\frac { 2 } { 7 }$
C)? $- \frac { 1 } { 3 }$
D)? $\frac { 21 } { 2 }$
E)?undefined

Question 44

Find the sum of the infinite geometric sequence.? $$\sum _ { n = 1 } ^ { \infty } 8 - \left( \frac { 1 } { 6 } \right) ^ { n - 1 }$$

A)?s = $\frac { 5 } { 34 }$
B)?s = $\frac { 34 } { 5 }$
C)?s = $\frac { 42 } { 5 }$
D)?s = $\frac { 46 } { 5 }$
E)?s = $\frac { 1 } { 5 }$

Question 45

Find the sum using the formulas for the sums of powers of integers. $$\sum _ { n = 1 } ^ { 8 } n ^ { 3 }$$

A)784
B)2,592
C)512
D)204
E)1296

Question 46

Find the sum of the indicated terms of the geometric sequence. ​
6,18,54 ...(to 8 terms)
​

A)s = 19,686
B)s = 19,692
C)s = 19,680
D)s = 19,674
E)none of the above

Question 47

Find the rational number representation of the repeating decimal.? $$0.\overline{859}$$

A)? $\frac { 859 } { 999 }$
B)? $\frac { 859 } { 9 }$
C)? $\frac { 859 } { 9999 }$
D)? $\frac { 85.9 } { 999 }$
E)? $\frac { 859 } { 99 }$

Question 48

Find the sum of the indicated terms of the geometric sequence.? $$\sum _ { n = 1 } ^ { n = 8 } 3 ( 4 ) ^ { n - 1 }$$ ?

A)s = 65,535
B)s = 65,538
C)s = 65,532
D)s = 65,541
E)s = 262,140

Question 49

?Find the sum of the finite geometric sequence. ?? $$\sum _ { n = 1 } ^ { 10 } \left( - \frac { 3 } { 4 } \right) ^ { n - 1 }$$ ?

A)? $\frac { 1,225,723 } { 262,144 }$
B)? $\frac { 141,361 } { 262,144 }$
C)? $\frac { 1,068,259 } { 262,144 }$
D)? $- \frac { 1,225,723 } { 262,144 }$
E)? $- \frac { 1,068,259 } { 262,144 }$

Question 50

Use a graphing utility to graph the first 10 terms of the sequence.? $$a _ { n } = 3 ( - 0.7 ) ^ { n - 1 }$$

A)? ?
B)? ?
C)? ?
D)? ?
E)

Question 51

Write the nth term of the geometric sequence as a function of n. $$a _ { 1 } = 2 , a _ { k + 1 } = 4 a _ { k }$$

A) $a _ { n } = 2 ( 4 ) ^ { n - 1 }$
B) $a _ { n } = - 2 + 4 n$
C) $a _ { n } = 2 ( 4 ) ^ { n }$
D) $a _ { n } = 2 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$
E) $a _ { n } = 4 ( 2 ) ^ { n - 1 }$

Question 52

Find Pk₊₁ for the given P_k. $$P _ { k } = \frac { 3 } { k ( k + 7 ) }$$

A) $P _ { k + 1 } = \frac { 3 } { k ( k + 7 ) } + 1$
B) $P _ { k + 1 } = \frac { 3 } { k ( k + 7 ) } + \frac { 3 } { ( k + 1 ) ( k + 8 ) }$
C) $P _ { k + 1 } = \frac { 3 } { ( k + 1 ) ( k + 8 ) }$
D) $P _ { k + 1 } = \frac { 3 } { k ( k + 8 ) }$
E) $P _ { k + 1 } = \frac { 8 } { ( k + 1 ) ( k + 7 ) }$

Question 53

Find the indicated nth term of the geometric sequence. $$10 \text { th term: } 2 , - 10,50 , \ldots$$

A)-5,120
B)-3,906,250
C)-2,560
D)-43
E)19,531,250

Question 54

Match the geometric sequence with its graph from the choices below.? $$a _ { n } = - 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }$$

A)? ?
B) ?
C)
D)
E)

Question 55

Find the sum of the finite geometric sequence.Round to the nearest thousandth. $$\sum _ { i = 0 } ^ { 5 } 600 ( 1.05 ) ^ { i }$$

A)4081.148
B)3197.686
C)4285.205
D)3481.148
E)3315.379

Question 56

Change the decimal to a common fraction.? $$0 . \overline { 5 }$$

A) $\frac { 5 } { 90 }$
B) $\frac { 5 } { 99 }$
C) $\frac { 5 } { 9999 }$
D) $\frac { 5 } { 9 }$
E) $\frac { 5 } { 999 }$

Question 57

Find the sum of the infinite geometric sequence. ​
2 + 0.4 + 0.08 ...
​

A)s = 2.5
B)s = 6.5
C)s = 0.1
D)s = 4.5
E)s = 0.5

Question 58

?Find the indicated nth term of the geometric sequence. ?
7th term: $$a _ { 4 } = - \frac { 4 } { 27 } , a _ { 10 } = - \frac { 4 } { 19,683 }$$ ?

A)? $$\frac { 4 } { 6,561 }$$
B)? $$- \frac { 4 } { 243 }$$
C)? $- \frac { 3 } { 4,096 }$
D)? $\frac { 4 } { 729 }$
E)? $- \frac { 4 } { 2,187 }$

Question 59

Use summation notation to write the sum. $$- 3 - 6 - 12 - \ldots - 48$$

A) $\sum _ { n = 1 } ^ { 5 } - 3 ( 2 ) ^ { n - 1 }$
B) $\sum _ { n = 1 } ^ { 4 } - 3 ( 2 ) ^ { n + 1 }$
C) $\sum _ { n = 0 } ^ { 4 } - 3 ( 2 ) ^ { n - 1 }$
D) $\sum _ { n = 1 } ^ { 3 } - 3 ( 2 ) ^ { n }$
E) $\sum _ { n = 1 } ^ { 3 } - 3 ( 2 ) ^ { n - 1 }$

Question 60

Find the eighth term of the geometric sequence whose second and fourth terms are 6 and 54. ​

A)a₈ = 39,366
B)a₈ = 4,374
C)a₈ = 13,122
D)a₈ = 1,458
E)a₈ = 486

Question 61

Prove the inequality for the indicated integer values of n.

Question 62

Find the sum using the formulas for the sums of powers of integers. $$\sum _ { n = 1 } ^ { 8 } \left( 7 n - 3 n ^ { 2 } \right)$$

A)-224
B)-1080
C)-136
D)144
E)-360

Question 63

Write the first four terms of the geometric sequence with the given properties.
​
a = 3,r = 2
​
Please enter your answer as four numbers in order,separated by commas.

Question 64

Find a quadratic model for the sequence with the indicated terms. $$a _ { 0 } = 3 , a _ { 1 } = 1 , a _ { 4 } = 31$$

A) $a _ { n } = 3 n ^ { 2 } + 3$
B) $a _ { n } = 3 n ^ { 2 } - 5 n + 3$
C) $a _ { n } = 3 n ^ { 2 } - 3 n + 3$
D) $a _ { n } = 3 n ^ { 2 } - 3 n + 5$
E) $a _ { n } = n ^ { 2 } + 5 n + 3$

Question 65

Use mathematical induction to prove the formula for every positive integer n.Show all your work.

Question 66

Find the sum of the indicated terms of the geometric sequence.​

Question 67

Use mathematical induction to prove the formula for every positive integer n.Show all your work.