For each of the following sequences:
Identify if the sequence is arithmetic or geometric.
Write the common ratio or difference.
11, -99, 891, -8019, \ldots
2, 6, 10, 14, \ldots
Determine whether each of the following is a geometric progression:
4 ,- 4, 4 ,- 4, \ldots
1,\sqrt{6}, 6, 6 \sqrt{6}, \ldots
2,0, - 2, - 4, \ldots
2, 2^{2}, 2^{4}, 2^{6}, \ldots
Suppose t_{1}, t_{2},t_{3},t_{4},t_{5}, \ldots is a geometric sequence.
Is t_{1}, t_{3}, t_{5}, \ldots a geometric sequence?
The first four terms of a geometric sequence are - 8,- 16 ,- 32,- 64. Evaluate:
\dfrac{T_2}{T_1}
\dfrac{T_3}{T_2}
\dfrac{T_4}{T_3}
T_5
Explain how the common ratio of a geometric sequence can be obtained.
Find the common ratio of the geometric sequence: - 70.4, - 17.6, -4.4, -1.1, \ldots
The first two terms of a geometric sequence are \sqrt{5} + \sqrt{3} and \sqrt{5} - \sqrt{3}.
Find the common ratio.
For each of the following, write the first four terms in the geometric progression:
The first term is 6 and the common ratio is 4.
The first term is 7 and the common ratio is - 2.
Write down the next two terms in the following sequences:
4, 12, 36
12, -48, 192
1, \dfrac{3}{4}, \dfrac{9}{16}
- 6, 9, - \dfrac{27}{2}
Find the next two terms in the sequence in terms of n:
n - 5,n^{2} + 5 n,n^{3} - 5 n^{2}, n^{4} + 5 n^{3}, \ldots
For each defined nth term of a sequence:
State the first four terms.
Use the common ratio to find the missing terms in the following geometric progressions:
- 5,ā¬,- 80, 320, ā¬
ā¬,ā¬,\dfrac{3}{25},- \dfrac{3}{125},ā¬
18, ā¬, ā¬, ā¬,\dfrac{32}{9}
Consider the sequence 5, - 5 , 5, - 5 , 5, - 5 , \ldots
Find the next term in the sequence.
Do the terms approach a single value as we progress along the sequence?
Consider the sequence: 9, 3, 1, \dfrac{1}{3}, \ldots
Find the next term in the sequence.
What value do the terms approach as we progress along the sequence?
Will there eventually be a term in the sequence that is equal to zero?
Consider the sequence: 6, 30, 150, 750, \ldots
Find the next term in the sequence.
What value do the terms approach as we progress along the sequence?
Will there eventually be a term in the sequence that is equal to infinity?
If t_n = 8 r^{n - 1}, determine if the following values of r would cause the terms to converge as n gets very large:
0.75
1.25
2^{-1}
\dfrac{9}{8}
Determine whether the terms in the following sequences are convergent or divergent:
3, 6, 12, 24, \ldots
60, 30, 15, \dfrac{15}{2}, \ldots
162, 54, 18, 6, \ldots
1, - 2 , 4, - 8 , \ldots
90, 30, 10, \dfrac{10}{3}, \ldots
6, 12, 24, 48, \ldots
t_n = 4 \times \left(0.5\right)^{n - 1}
t_n = 4 \times 3^{n - 1}
t_n = 0.25 \times 4^{n - 1}
t_n = 4 \times \left(\dfrac{2}{3}\right)^{n - 1}
t_n = 5 \left(\dfrac{- 2}{3}\right)^{n - 1}
t_n = 0.5 \times 2^{n - 1}
t_n = 5 \times \left(\dfrac{5}{4}\right)^{n - 1}
t_n = 2 \left( - 0.75 \right)^{n - 1}
Consider the following sequences:
Find the next term.
Find the common ratio, r.
Write an expression for the general nth term of the sequence, t_n.
What value do the terms approach as n gets very large?
Consider the sequence: - 2.5 , 5, - 10 , 20, \ldots
Find the next term in the sequence.
Determine the common ratio.
Write an expression for the general nth term of the sequence t_n.
Do the terms converge or diverge as the sequence progresses?
Three consecutive positive terms of a geometric progression have a product of 125. The third term is 9 times the first.
Find the middle term.
Find the three consecutive terms.
Find r, the common ratio.
Consider the following finite sequences:
Find the common ratio.
Find T_6.
Find n, the number of terms.
In a geometric progression, T_7 = \dfrac{64}{81} and T_8 = \dfrac{128}{243}.
Find the value of r, the common ratio in the sequence.
Find the first three terms of the geometric progression.
Consider the following sequence: - 0.3, -1.5, -7.5,-37.5, \ldots
Find the formula for the nth term of the sequence.
Hence, find the next three terms of the sequence.
Consider the following sequence: 2,- 16, 128, - 1024, \ldots
State the general expression for the nth term of the sequence.
Hence, find the next three terms of the sequence.
Find T_{11}.
Consider the following:
1, \, x and y are the first three terms of an arithmetic sequence. Form an equation for y in terms of x.
1, \, y and x are also the first three terms in a geometric sequence. Form an equation for x in terms of y.
Hence, find the value(s) of y.
When y = 1 and x = 1 it produces the sequence 1, 1, \, 1.
Find the first three values of the arithmetic sequence for the other solution for x and y, along with the common difference.
Find the first three values of the geometric sequence for the other solution for x and y, along with their common ratio.
Consider the sequence: 9000, 1800, 360, 72, \ldots
Write a recursive rule for T_{n+1} in terms of T_n and an initial condition for T_1.
The first term of a geometric sequence is 6. The fourth term is 384.
Find the common ratio, r.
Write the recursive rule, T_{n+1}, that defines this sequence.
For each of the following:
Find the value of the common ratio, r.
Find the first term, T_1.
Write a recursive rule, T_{n+1}, that defines the sequence with a positive common ratio.
Write a recursive rule, T_{n+1}, that defines the sequence with a negative common ratio.
The third term of a geometric sequence is 2500. The seventh term is 4.
The first term of a geometric sequence is 5. The third term is 80.
In a geometric progression, T_4 = - 192 and T_7 = 12\,288.
Find the value of r, the common ratio in the sequence.
Find T_1, the first term in the progression.
Find an expression for T_n, the general nth term.
In a geometric progression, T_4 = 32 and T_6 = 128.
Find r, the common ratio in the sequence.
For the case where r is positive, find T_1, the first term in the progression.
Given that T_1 is positive, find an expression for T_n, the general nth term of this sequence.
Consider the sequence: 54,18, 6, 2, \ldots
Plot the first four terms on a graph
Is relationship depicted by the graph linear, exponential or neither?
As n approaches infinity, which value does the nth term approach?
Consider the first-order recurrence relationship defined by T_{n+1} = 2T_n, T_1 = 2.
Determine the next three terms of the sequence from T_2 to T_4.
Plot the first four terms on a graph.
Is the sequence generated from this definition arithmetic or geometric?
Consider the following sequences:
Plot the first four terms on a graph.
Is the relationship depicted by the graph linear, exponential, or neither?
State the recurrence relationship, T_{n+1}, in terms of T_n, that defines the sequence.
Consider the sequence plot below:
State the terms of the first five points of the sequence.
Is the relationship depicted by the graph linear, exponential, or neither?
As n approaches infinity, do the terms t_n converge or diverge?
Consider the sequence plot below:
State the terms of the first five points of the sequence, from T_1 to T_5.
Is the sequence depicted by this graph arithmetic or geometric? Explain your answer.
Write a recursive rule for T_{n+1} in terms of T_n and an initial condition for T_1.
Consider the sequence plot drawn below:
State the terms of the first five points of the sequence.
Is the relationship depicted by this graph linear, exponential, or neither?
State the recurrence relationship, T_{n+1}, that defines this sequence.
Complete the missing values in the table:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | -27 | -64 |
The nth term of a geometric progression is given by the equation T_n = 2 \times 3^{n - 1}.
Complete the table of values:
Find the common ratio between consecutive terms.
Plot the first four points on a graph.
If the plots on the graph were joined, do they form a straight or a curve?
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
The nth term of a geometric progression is given by the equation T_n = 25 \times \left(\dfrac{1}{5}\right)^{n - 1}.
Complete the table of values.
Find the common ratio between consecutive terms.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
The given table of values represents terms in a geometric sequence.
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 12th term of the sequence.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 7 | -21 | 63 | -189 |
The given table of values represents terms in a geometric sequence.
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 10th term of the sequence.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 5 | 40 | 320 | 2560 |
The given table of values represents terms in a geometric sequence.
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 7th term of the sequence. Round your answer to three decimal places.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | -2 | -\dfrac{16}{3} | -\dfrac{128}{9} | -\dfrac{1024}{27} |
For each set of plotted points that represent terms in a geometric sequence:
Create a table of values for the given points.
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 10th term of the sequence.
The plotted points represent terms in a geometric sequence:
Find r, the common ratio.
Write a simplified expression for the general nth term of the sequence, T_n.
The points are reflected about the horizontal axis to form three new points.
If these new points represent consecutive terms of a geometric sequence, write the equation for T_k, the kth term in this new sequence.
The 6th term of a geometric sequence is 557 and the 13th term is 255\,642.
Write an equation involving T_1, the first term, and r, the common ratio, for the 6th term.
Write an equation involving T_1, the first term, and r, the common ratio, for the 13th term.
Give the value of T_1 to the nearest integer and give the value of r to one decimal place. Assume T_1 and r are positive.
Hence, determine the 8th term of the sequence.
The 7th term of a geometric sequence is 353 and the 13th term is 42.
Write an equation involving T_1, the first term, and r, the common ratio, of this geometric sequence for the 7th term.
Write an equation involving T_1, the first term, and r, the common ratio, of this geometric sequence for the 13th term.
Use the simultaneous solving facility of your calculator to find the values of T_1 and r.
Give the value of T_1 to the nearest integer and give the value of r to one decimal place. Assume T_1 and r are positive.
Hence, determine the 9th term of the series.
The 5th term of a geometric sequence is 11 and the 12th term is 72.
Write an equation involving T_1, the first term, and r, the common ratio, for the 5th term.
Write an equation involving T_1, the first term, and r, the common ratio, for the 12th term.
Use the simultaneous solving facility of your calculator to find the values of T_1 and r.
Give the value of T_1 to the nearest integer and give the value of r to one decimal place. Assume T_1 and r are positive.
Hence, state the general term, T_n, of the sequence.
The 4th term of a geometric sequence is 33 and the 14th term is 952.
Write an equation involving T_1, the first term, and r, the common ratio, for the 4th term.
Write an equation involving T_1, the first term, and r, the common ratio, for the 14th term.
Give the value of T_1 to the nearest integer and give the value of r to one decimal place in the general term. Assume T_1 and r are positive.
The first three terms of a geometric sequence are x + 11, x + 2 and x - 4.
Form an equation in terms of x and then use a graphing calculator to find x.
Hence, find the common ratio of the sequence.
Hence, determine the 4th term of the sequence.
For each of the following investments:
Find how much money is in the account at the end of the first year.
Write a recursive rule, V_{n+1}, that gives the balance in the account at the end of year n+1.
\$2000 is invested at the beginning of the year in an account that earns 4\% per annum interest, compounded quarterly.
\$4000 is invested at the beginning of the year in an account that earns 8\% per annum interest, compounded quarterly.
\$5000 is invested at the beginning of the year in an account that earns 6\% per annum interest, compounded monthly.
\$3000 is invested at the beginning of the year in an account that earns 6\% per annum interest, compounded quarterly.
The balance of an investment at the end of the nth year where interest is compounded annually is given by A_{n+1} = 1.05 A_{n}, A_0 = 30\,000.
State the annual interest rate.
State the amount invested.
Determine the balance at the end of the first year.
Calculate the balance at the end of 15 years.
The balance of an investment at the end of the nth year where interest is compounded annually is given by A_{n+1} = 1.061 A_{n}, A_0 = 15\,000.
State the annual interest rate.
State the amount invested.
Determine the balance at the end of the first year.
Determine the balance at the end of 20 years.
The balance of an investment at the beginning of each quarter where interest is compounded quarterly is given by A_{n+1} = 1.02 A_{n}, A_1 = 5000.
State the quarterly interest rate.
State the nominal annual interest rate.
Determine the balance at the beginning of the second year.
Determine the balance at the end of the second year.
The balance of an investment at the end of each month where interest is compounded monthly is given by A_{n+1} = 1.015 A_{n}, A_0 = 2000. The investment began at the beginning of January 2010.
State the monthly interest rate.
Use the sequences facility on your calculator to determine the balance at the end of the first year.
Use the compound interest formula to determine the balance at the end of the first year to confirm the answer from the previous part.
Determine in which month and year the investment is worth double the initial amount invested.
\$3900 is invested for three years at a rate of 10\% per annum, compounding annually. The balance and interest for the first two years are shown in the table:
Write the recurrence relation for this situation.
Use the sequence facility of your calculator to find the value of:
Calculate the total interest earned over the three years.
| Balance | Interest earned | |
|---|---|---|
| First year | \$3,900 | \$390 |
| Second year | \$4,290 | \$429 |
| Third year | \$A | \$B |
| Fourth year | \$C | - |
\$8000 is invested at 6\% p.a., compounded annually.
Write the recurrence relation for this situation.
Using the sequence facility of your calculator, complete the following table:
| Time Period (years) | Value at beginning of time period | Value at end of time period | Interest earned in time period |
|---|---|---|---|
| 1 | \$8,000 | ||
| 2 | \$8,988.80 | ||
| 3 | \$8,988.80 | \$9,528.13 |
Find the total interest earned over the three years.
Callum invests \$5700 into an investment account that pays 3.2\% per annum, compounded annually.
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write an explicit rule that can be used to find the balance at the end of n years.
Use the sequences application on your calculator to determine the balance after 9 years.
Determine how many whole years it takes for the balance to exceed \$10\,208.
Erica invests \$50\,000 into an investment account that pays 2.8\% per annum, compounded annually.
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write an explicit rule that can be used to find the balance at the end of n years.
Use the sequences application on your calculator to determine the balance after 21 years.
Determine how many whole years it takes for the balance to exceed \$109\,843.
Juan invests \$25\,000 into an investment account that pays 1.8\% compound interest per annum, compounded quarterly.
Find the quarterly interest rate.
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth quarter and t_0 is the initial investment.
Write an explicit rule that can be used to find the balance at the end of n quarters.
Use the sequences application on your calculator to determine the balance after 6 years.
Determine how many whole years it takes for the balance to exceed \$28\,800.
Rani invests 27\,500 INR into an investment account that pays 6.3\% compound interest per annum, compounded monthly.
Find the monthly interest rate.
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth month and t_0 is the initial investment.
Write an explicit rule that can be used to find the balance at the end of n months.
Use the sequences application on your calculator to determine the balance after 4 years.
Determine how many whole months it takes for the balance to exceed 35\,543 INR.
Convert your answer from part (e) to years. Round your answer to two decimal places.
Briony invests \$29\,000 into an investment account that pays 2.2\% compound interest per annum.
Complete the recurrence relation for this situation, where t_n describes the value of the investment after the nth year.
Write an explicit rule that can be used to find the balance at the end of n years.
Use the sequences application on your calculator to determine the balance after 5 years.
Determine how many whole years it takes for the balance to exceed \$41\,527.
The following table shows the mass of 800 grams of radioactive element D left each day, given that element D loses half its mass every day.
Complete the table.
What type of decay is this, linear or exponential?
| Day | Mass of element D (g) |
|---|---|
| 0 | 800 |
| 1 | |
| 2 | |
| 3 | |
| 4 |
The average daily growth of a seedling is 6\% per day. A seedling measuring 8 \text{ cm} in height is planted.
Find the height of the seedling at the end of day 1.
Find the height of the seedling 4 days after it is planted
Write a recursive rule, H_{n+1} in terms of H_n, defining the height of the seedling n+1 days after it is planted.
The average annual rate of inflation in Kazakhstan is 2.6\%. Bread cost \$3.65 in 2015.
Find the cost of bread in 2016.
At this rate, find the bread cost in 2018.
Write a recursive rule, V_{n+1}, defining the cost of bread n+1 years after 2015.
The zoom function in a camera multiplies the dimensions of an image. In an image, the height of waterfall is 30\text{ mm}. After the zoom function is applied once, the height of the waterfall in the image is 36 \text{ mm}. After a second application, its height is 43.2 \text{ mm}.
Each time the zoom function is applied, by what factor is the image enlarged?
If the zoom function is applied a third time, find the exact height of the waterfall in the image.
Suppose you save \$1 the first day of a month, \$2 the second day, \$4 the third day, \$8 the fourth day, and so on. That is, each day you save twice as much as you did the day before.
How much will you save on the 17th day of the month?
How much will you save on the 29th day of the month?
A car enthusiast purchases a vintage car for \$220\,000. Each year, its value increases at a rate of 12 percent of its value at the beginning of the year.
Find its value after 7 years, to two decimal places.
A gym trainer posts Monday's training program on the board, along with how you should progress each day that follows based on your level of fitness:
\enspace
MONDAY TRAINING PROGRAM
| Single rope skips | 9 |
|---|---|
| Weight lift | 6 \dfrac{1}{2} \text{ kg} |
| Rest | 2 \text{ minutes} |
| Row | \dfrac{1}{4} \text{ mile} |
BEGINNER LEVEL:
Each day, increase the numbers and time by \dfrac{1}{3} of the first day.
INTERMEDIATE LEVEL:
Each day, increase the numbers and time by \dfrac{1}{3} of the previous day.
Using the Intermediate Level training program:
Find the number of single rope skips you would need to complete on Wednesday.
Find the weight you would need to weight lift on Wednesday as a mixed number.
Find the rest time on Wednesday.
Find the distance to be rowed on Wednesday.
Using the Beginner Level training program:
Find the number of single rope skips that will need to be done on Wednesday.
Find the distance you would need to row on Wednesday.
Which level training plan is the most realistic in the long term, Beginner or Intermediate?
A sample of 2600 bacteria was taken to see how rapidly the bacteria would spread. After 1 day, the number of bacteria was found to be 2912.
By what percentage had the number of bacteria increased over a period of one day?
If the bacteria continue to multiply at this rate each day, what will the number of bacteria grow to eighteen days after the sample was taken? Round to the nearest whole number.
A rectangular poster originally measures 81 centimetres in width and 256 centimetres in length. To edit the poster once, the length of the rectangle is decreased by \dfrac {1}{4} and the width is increased by \dfrac {1}{3}.
If the poster is edited once, find the ratio of the original area of the rectangle to the new area.
If the edit is repeated 3 times, find the new area of the poster to the nearest square centimetre.
Find the number of times, n, that the process must be repeated to produce a square poster.
To test the effectiveness of a new antibiotic, a certain bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are 19 bacteria in the body, and after four hours, the number is found to double.
If the bacterial population continues to double every four hours, how many bacteria will there be in the body after 24 hours?
The antibiotic is applied after 24 hours, and is found to kill one third of the germs every two hours.
How many bacteria will there be left in the body 24 hours after applying the antibiotic? Assume the bacteria stops multiplying and round your answer to the nearest integer if necessary.