Deck 25: Sequences, Series, and the Binomial Theorem

Question 1

Write the first five terms of the following series, given the general term: $U_{n}=3 n^{2}$

Answer 1

$3+12+27+48+75+\ldots$

Question 2

Write the first five terms of the series that has the general term: $U_{n}=(2-n)^{2}$ .

Answer 2

$1+0+1+4+9+\ldots$

Question 3

Write the first five terms of the series:
$U_{n}=\frac{n+2}{n+3}$

Answer 3

$\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}+\frac{7}{8}+\ldots$

Question 4

Write the first five terms of the series: $U_{n}=\frac{n-1}{n+4}$

Question 5

Write the first five terms of the series: $U_{n}=2^{n}-1$

Question 6

Write the first five terms of the series:
$U_{n}=\frac{n+1}{n^{2}-2}$

Question 7

Deduce the general term of the series: $1+\frac{4}{3}+\frac{5}{3}+\ldots$

Question 8

Deduce the general term of the series: $0+\frac{1}{4}+\frac{2}{5}+$

Question 9

Deduce the general term of the series: $6+12+20+\ldots$ Use the general term to predict the next two terms.

Question 10

Deduce the general term of the series: $3+8+13+\ldots$ Use the general term to predict the next two terms.

Question 11

Deduce the general term of the series: $-1-3-5-\ldots$ Use the general term to predict the next two terms.

Question 12

Deduce the general term of the series: - $2-6-10-\ldots$ Use the general term to predict the next two terms.

Question 13

Deduce a recursion relation for the series: $6+12+24+\ldots$ Use it term to predict the next two terms.

Question 14

Deduce a recursion relation for the series: $2+4+12+48+\ldots$ Use it term to predict the next two terms

Question 15

Deduce the general term for the series: $0+3+8+15+24+\ldots$ Use the general term to predict the next two terms.

Question 16

Deduce the general term for the series: $2+4+16+\ldots$ Use the general term to predict the next two terms.

Question 17

Write the first five terms of the following series, given the general term: $U_{n}=\frac{2 n+1}{n^{2}}$

Question 18

Deduce the general term of the series: $0+4+12+24+\ldots$ Use the general term to predict the next two terms.

Question 19

Deduce a recursion relation for the series: $1+5+17+\ldots$ Use it to predict the next two terms.

Question 20

Find the seventh term of an AP with first term 3 and common difference 5 .

Question 21

Find the tenth term of an AP with first term 2 and common difference 7.

Question 22

Find the sixteenth term of an AP with first term 5 and seventh term 23.

Question 23

Find the twelfth term of the AP: $8,22,36, \ldots$

Question 24

Find the ninth term of the AP: $a, a+6 b, a+12 b, \ldots$

Question 25

Find the ninth term of the AP: $a, 2 a-b, 3 a-2 b, \ldots$

Question 26

Find the tenth term of the AP: $2 \frac{1}{2}, 4,5 \frac{1}{2}, \ldots$

Question 27

Find the tenth term of the AP: $\frac{1}{2}, 0,-\frac{1}{2},-1,-1 \frac{1}{2}, \ldots$

Question 28

Write the first five terms of the following AP: first term is -2 and tenth term is 25

Question 29

Write the first five terms of the following AP: first term is 3 and eighth term is 24

Question 30

Find the first term of an AP whose common difference is -2 and whose ninth term is 27 .

Question 31

Find the first term of an AP whose common difference is 4 and whose tenth term is 45 .

Question 32

Find the sum of the first 8 terms of an AP with first term 6 and difference 4 .

Question 33

Find the sum of the first 9 terms of the following AP: $-2,2,6, \ldots$

Question 34

Find the sum of the first 18 terms of the AP: $2,7,12,17, \ldots$

Question 35

Find the sum of the first 25 terms of the AP: $-12,-7,-2,3,8, \ldots$

Question 36

Find the sum of the first 12 terms of the AP: $7,16,25,34,43, \ldots$

Question 37

How many terms of the AP $-1,1,3, \ldots$ will give the sum 399 ?

Question 38

How many terms of the AP $1,5,9,13,17, \ldots$ will give the sum 2016 ?

Question 39

Insert four arithmetic means between 20 and 23.

Question 40

Insert five arithmetic means between -3 and 21 .

Question 41

Insert three arithmetic means between 2 and 66.

Question 42

Find the fifteenth term of the following harmonic progression: $\frac{7}{39}, \frac{7}{33}, \frac{7}{27}, \ldots$

Question 43

Find the tenth term of the following harmonic progression: $\frac{17}{2}, \frac{17}{19}, \frac{17}{36}, \ldots$

Question 44

Find the eighth term of the following harmonic progression: $\frac{8}{35}, \frac{8}{27}, \frac{8}{19}, \ldots$

Question 45

Insert 4 harmonic means between $\frac{8}{11}$ and $\frac{8}{31}$ .

Question 46

Insert 12 harmonic means between $\frac{1}{17}$ and $\frac{1}{82}$ .

Question 47

Insert 3 harmonic means between $\frac{3}{4}$ and $\frac{3}{16}$ .

Question 48

A person is hired at a salary of $\$ 45,000$ and receives a raise of $\$ 3,200$ at the end of each year. Find the total amount earned during 20 years.

Question 49

A certain milling machine has an initial value of $\$ 300,000$ and a scrap value of $\$ 20,000$ fourteen years later. Assuming that the machine depreciates the same amount each year, find its value after eight years.

Question 50

Find the sum of the first 18 terms of the AP: $1,9,17,25,33, \ldots$

Question 51

Insert 3 harmonic means between $\frac{3}{5}$ and $\frac{3}{29}$ .

Question 52

Find the seventh term of a GP with first term 4 and common ratio -3 .

Question 53

Find the tenth term of a GP with first term -5 and common ratio 3 .

Question 54

Find the tenth term of a GP with first term 4 and common ratio -2 .

Question 55

Find the seventh term of a GP with first term 3 and common ratio 7 .

Question 56

Find the eighth term of a GP with first term 2 and common ratio -5 .

Question 57

Find the sum of the first eight terms of a GP with first term 5 and common ratio 2 .

Question 58

Find the sum of the first seven terms of the following GP: $1,5,125,625, \ldots$

Question 59

Find the sum of the first ten terms of the following GP: $4,-12,36, \ldots$

Question 60

Find the sum of the first six terms of the following GP: $8,6,4 \frac{1}{2}, 3 \frac{3}{8}, \ldots$

Question 61

Find the sum of the first six terms of the following GP: $-2,1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8} \ldots$

Question 62

Insert a geometric mean between -8 and -512 .

Question 63

Insert two geometric means between 7 and 56 .

Question 64

Insert two geometric means between 3 and 192 .

Question 65

Insert two geometric means between 6 and 2058.

Question 66

Insert two geometric means between 47 and 1269.

Question 67

Insert three geometric means between 31 and 496.

Question 68

A certain radioactive material decays so that after each year the radioactivity is $8 \%$ less than at the start of that year. How many years will it take for its radioactivity to be $10 \%$ of its original value? Use three significant digits.

Question 69

Each day the size of a certain colony of bacteria is $20 \%$ larger than on the preceding day. If the original size of colony was 25000 bacteria, find its size after 6 days.

Question 70

A certain metal casting is at $900^{\circ} \mathrm{C}$ and is cooling so that each minute its temperature is $7.5 \%$ less than the previous minute. Find its temperature after 30 minutes? Use three significant digits.

Question 71

The sum of the first and third term of a GP is 15 , and the sum of the second and fourth term is 30 . Find the sum of the first ten terms of this GP.

Question 72

The difference between the second and fourth term of a GP is 24 , and the difference between the fifth and seventh term is 648 . Find the sum of the first six terms of this GP.

Question 73

The first term of a GP is 1 , and the sum of the third and fifth term is 90 . Determine the first five terms of this GP.

Question 74

The first term of a GP is 3 , and the difference between the seventh and fourth term is 168 . Determine the first five terms of this GP. Assume that the common ratio is positive.

Question 75

Find the tenth term of a GP with first term 5 and common ratio 2.

Question 76

Find the sum of the first twelve terms of a GP with first term -3 and common ratio -2 .

Question 77

$\$ 8000$ is invested in an account with a $6 \%$ interest rate compounded monthly. What is the value, to the nearest dollar, of the account after 5 years?

Question 78

Evaluate the limit: $\lim_{b \rightarrow 0} \frac{2+b^{2}}{a+c}$

Question 79

Evaluate: $\lim_{b \rightarrow 0} \frac{a+b-c}{b-a+c}$

Question 80

Evaluate: $\lim_{b \rightarrow 0} \frac{4 b-2 c}{5-b}$