Explain how the common ratio of a geometric sequence can be found.
Determine whether the following are geometric sequences:
1, \sqrt{6}, 6, 6 \sqrt{6} , \ldots
3, 1, -1, -3, \ldots
4, -4, 4, -4, \ldots
3, 3^{2}, 3^{4}, 3^{6}, \ldots
Consider the following sequence: 2, 6, 10, 14, \ldots
Determine if the sequence is arithmetic or geometric. Explain your answer.
Find the common ratio or common difference, whichever is applicable.
Consider the sequence: 4, - 28, 224, - 1372, \ldots
Is the sequence arithmetic, geometric or neither?
Suppose t_{1}, t_{2}, t_{3}, t_{4}, t_{5},\ldots is a geometric sequence. Is t_{1}, t_{3}, t_{5},\ldots also a geometric sequence? Explain your answer.
Write down the next two terms for the following sequences:
4, 12, 36, \ldots
12, -48, 192, \ldots
1, \dfrac {3}{4}, \dfrac {9}{16}, \ldots
- 6, 9, - \dfrac {27}{2}, \ldots
n - 5, n^{2} + 5 n, n^{3} - 5 n^{2}, n^{4} + 5 n^{3}, \ldots
Write down the first four terms in the following geometric sequences:
The first term is - 6 and the common ratio is 4.
The first term is - 5 and the common ratio is - 4.
The first term is 700\,000 and the common ratio is 1.04.
The first term of 1.3 and a common ratio of - 4.
If the first term of a sequence is 1.2 and the common ratio is - 5, find the 4th term.
If the first term of a sequence is 400\,000 and the common ratio is 1.12, find the 3rd term.
Find the common ratio of the following geometric sequences:
2, - 16, 128, - 1024, \ldots
- 70.4, - 17.6, - 4.4, - 1.1, \ldots
Consider the sequence: -1, -7, -49, \ldots
Find the next term of the sequence.
Find the 5th term of the sequence.
Find the 6th term of the sequence.
Consider the first four terms in this geometric sequence: - 8, - 16, - 32, - 64,\ldots
Evaluate:
\dfrac {T_2}{T_1}
\dfrac {T_3}{T_2}
\dfrac {T_4}{T_3}
Hence, find the value of T_5.
Find the common ratio for the geometric sequence where the first two terms are \sqrt{5} + \sqrt{3} and \sqrt{5} - \sqrt{3}.
Use the common ratio to find the missing terms in the following geometric sequences:
- 9, ⬚, - 144, - 576, ⬚
⬚, ⬚, - \dfrac{5}{16}, - \dfrac{5}{64}, ⬚
Find the missing values between 12 and \dfrac{243}{4} such that the 5 terms form the successive terms in a geometric sequence:
12 , ⬚ , ⬚ , ⬚ , \dfrac{243}{4}
The first term of a geometric sequence is 5 and the common ratio of the sequence is 3.
Find the 4th term of the sequence.
Find the first term of the sequence that is greater than a million.
Is the number 80\,700\,215 a term of the sequence?
The first term of a geometric sequnce is 800 and the common ratio is \dfrac{1}{4}.
Is 8 a term of the sequence?
Find the first term less than \dfrac{1}{10}.
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.
Given the nth term of the following sequences defined by the given equation:
Write the first four terms of the sequence.
Find the common ratio.
T_n = 3 \times 4^{n - 1}
T_n = - 4 \times \left( - 3 \right)^{n - 1}
Find n, the number of terms in the geometric sequence: 162, 54, 18 , \ldots , \dfrac{2}{9}
Consider the finite sequence: 4, 4 \sqrt{2}, 8, \ldots, 256
Find the common ratio.
Find T_6.
Find n, the number of terms in the sequence.
In a geometric sequence, T_{7} = \dfrac{64}{27} and T_{8} = \dfrac{128}{81}.
Find r, the common ratio in the sequence.
Find the first three terms of the geometric sequence.
In a geometric sequence, T_{4} = 320 and T_{8} = 81\,920.
Find r, the common ratio in the sequence.
For the case where r is positive, find a, the first term in the sequence.
Find an expression for T_{n}, the general nth term of this sequence.
In a geometric sequence, T_{3} = 64 and T_{6} = - 4096.
Find r, the common ratio in the sequence.
Find a, the first term in the sequence.
Find an expression for T_{n}, the general nth term.
Find three consecutive positive terms of a geometric progression if they have a product of 125 and the third term is 4 times the first.
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 values of y.
One solution is y = 1 and x = 1 which 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 the common ratio.
The third term of a geometric sequence is 8. The sixth term is 64.
Find the common ratio, r.
Find the first term of this sequence.
Find the recursive rule, T_{n}, that defines this sequence.
The first term of a geometric sequence is 6. The fourth term is 384.
Find the common ratio, r.
Find the recursive rule, T_{n}, that defines this sequence.
The 3rd term of a geometric sequence is 8100. The 7th term is 100.
Find the common ratio, r.
Find the first term of this sequence.
Find the recursive rule, T_{n}, that defines the sequence with a positive common ratio.
Find the recursive rule, T_{n}, that defines the sequence with a negative common ratio.
The first term of a geometric sequence is 5. The third term is 80.
Find the possible values of the common ratio, r.
Find the recursive rule, T_{n}, that defines the sequence with a positive common ratio.
Find the recursive rule, T_{n}, that defines the sequence with a negative common ratio.
For the sequence: 108, 18 , 3, \ldots
Find an expression for T_{n}.
Using logarithm laws, find how many terms of the sequence exceed 10^{-5}.
The sum of the 3rd and 5th terms of a geometric sequence is 20. The sum of the 2nd and 4th terms is 60.
Find the value of r.
Hence, find the value of a.
Find the first term that is less than 5.
Consider the sequence: 9000, 1800, 360, 72, \ldots
Write a recursive rule for T_n in terms of T_{n - 1} and an initial condition for T_1.
The nth term of a geometric sequence is given by the equation T_{n} = 2 \times 3^{n - 1}.
Complete the table of values:
Find the common ratio between consecutive terms.
| n | 1 | 2 | 3 | 4 | 11 |
|---|---|---|---|---|---|
| T_{n} |
Plot the points in the table that correspond to n = 1, 2, 3 and 4.
If the plots on the graph were joined together, will they form a straight line or an exponential curve?
The nth term of a geometric sequence is given by the equation T_{n} = 4 \times \left(\dfrac{1}{2}\right)^{n - 1}.
Complete the table of values:
Find the common ratio.
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_{n} |
For each of the following tables that represent terms in a geometric sequence:
Find r, the common ratio.
Write an expression for the nth term of the sequence, T_n.
Find the missing term in the table.
| n | 1 | 2 | 3 | 4 | 8 |
|---|---|---|---|---|---|
| T_n | 9 | 63 | 441 | 3087 |
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | - 4 | - \dfrac{40}{3} | - \dfrac{400}{9} | - \dfrac{4000}{27} |
The given table of values represents terms in a geometric sequence:
Find r, the common ratio.
Write an expression for the general nth term of the sequence, T_n.
| n | 1 | 5 | 8 |
|---|---|---|---|
| T_n | 2 | 4802 | - 1\,647\,086 |
The plotted points represent terms in a geometric sequence:
Complete the table of values for the given points:
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n |
Find r, the common ratio.
Write an expression for the general nth term of the sequence, T_{n}.
Find the 13th term of the sequence.
The plotted points represent terms in a geometric sequence:
Complete the table of values for the given points:
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n |
Find r, the common ratio.
Write an 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:
State the first term in the sequence.
Find r, the common ratio between consecutive terms.
Write an expression for the general nth term of the sequence, T_{n}.
The plotted points represent terms in a geometric sequence:
Find r, the common ratio between consecutive terms.
Write an expression for the general nth term of the sequence, {T_n}.
The points are reflected across 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 values in the table show the nth term in a geometric sequence for consecutive values of n:
Complete the table.
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_{n} | 4 | - 32 |
Consider the first-order recurrence relationship defined by T_n = 2 T_{n - 1},\text{ } T_1 = 2.
Determine the next three terms of the sequence from T_2 to T_4.
Plot the first four terms on a number plane.
Is the sequence arithmetic or geometric? Explain your answer.
Consider the sequence: 2, 6, 18, 54, \ldots
Plot the first four terms on a graph.
Is this sequence arithmetic or geometric?
Write a recursive rule for T_{n} in terms of T_{n - 1} 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 sequence arithmetic or geometric?
Write a recursive rule for T_{n} in terms of T_{n - 1} 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}, that defines this sequence.
Caitlin invests \$21\,000 into an account that earns 7\% p.a. with interest calculated at the end of each year.
Complete the following table. Round all values to the nearest dollar.
Find r, the common ratio between consecutive values of the investment from year to year.
| n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| A \, (\$) | 21\,000 |
Write an equation that models the value of the investment, A, after n years.
A new car purchased for \$38\,200 depreciates at a rate, r , each year. The value of the car over the first three years is shown in the following table:
| \text{Years passed } (n) | 0 | 1 | 2 |
|---|---|---|---|
| \text{Value of car } (A) | 38\,200 | 37\,818 | 37\,439.82 |
Use the table of values to determine the value of r.
Determine the rule for A, the value of the car, n years after it is purchased.
Assuming the rate of depreciation remains constant, how much can the car be sold for after 6 years?
A new motorbike purchased for the same amount depreciates according to the model \\ V = 38200 \times 0.97^{n}. Which vehicle depreciates more rapidly?
Joanne wants to invest \$1100 at 3\% p.a for 3 years. She has 2 investment options, compounding quarterly or compounding monthly, and wants to find the difference in the final investment values of these 2 options.
Calculate the value of the investment if it is compounded quarterly.
Calculate the value of the investment if it is compounded monthly.
Hence, calculate how much extra the investment is worth if it is compounded monthly rather than quarterly.
A \$9450 investment earns interest at 2.6\% p.a. compounded monthly over 14 years. Find the future value of the investment.
Ursula has just won \$30\,000. She decides to invest some of her winnings into a retirement fund which earns 8\% interest p.a., compounded yearly. When she retires in 29 years, she wants to have \$52\,000 in her fund.
How much of her winnings should Ursula invest now to achieve this?
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 put aside for savings on the 17th day of the month?
How much will you put aside for savings on the 29th day of the month?
A car enthusiast purchases a vintage car for \$220\,000. Each year, its value increases by 12 \% of its value at the beginning of the year.
Find the value of the car after 7 years.
The average rate of depreciation of the value of a Ferrari is 14\% per year. A new Ferrari is bought for \$90\,000.
Calculate the value of the car after 1 year.
Calculate the value of the car after 3 years.
Write a recursive rule, V_{n}, defining the value of the car after n years.
The average annual rate of inflation in Kazakhstan is 2.6\%. Bread cost \$3.65 in 2015.
What would the bread cost in 2016?
At this rate, what would bread cost in 2018?
Write a recursive rule, V_n, defining the cost of bread n years after 2015.
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 your answer to the nearest whole number.
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, what will be the exact height of the waterfall in the image?
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, to two decimal places, 4 days after it is planted.
Write a recursive rule, H_{n}, defining the height of the seedling n days after it is planted.
To test the effectiveness of a new antibiotic, first 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.
A rectangular poster originally measures 81 \text{ cm} in width and 256 \text{ cm} 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, calculate 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.
A deposit of \$5000 is made on June 1, 2006 into an investment account and a deposit of \$400 is made each year on May 31.
The balance at the end of each 12-month period for this investment, where interest is compounded annually, is given by A_{n} = 1.05 A_{n - 1} + 400, and A_{0} = 5000.
State the annual interest rate.
Find the balance on June 1, 2007.
Find the value of the investment on June 1, 2013.
Mr Smith opened a bank account for his granddaughter Avril on the day she was born, January 5, 2006. He deposited \$4000 into the account.
Mrs Smith, Avril’s grandmother, deposited \$400 into this account on that day, and continues to do so by depositing \$400 every 3 months.
The balance at the end of each quarter for this investment, where interest is compounded quarterly, is given by A_n = 1.03 A_{n - 1} + 400, A_0 = 4400.
State the quarterly interest rate.
Find the nominal annual interest rate.
Find the balance on the day after Avril's first birthday.
Find the balance on the day after Avril's 12th birthday.
Bill opens an account to help save for a house. He opens the account at the beginning of 2013 with an initial deposit of \$40\,000 that is compounded annually at a rate of 3.7\% per annum. He makes further deposits of \$1000 at the end of each year.
How much money is in the account at the end of the first year?
Write a recursive rule for V_n in terms of V_{n - 1} and an initial condition V_0, that gives the value of the account after n years.
Calculate the total value of his savings at the beginning of 2021.
Hence, determine how much interest Bill earned.
Sandy opens a savings account to motivate herself to save regularly. She opens the account at the start of September, 2013 with the intention of making regular deposits of \$110 at the end of each month. The interest rate for this account is 24\% per annum which compounds at the end of each month.
Find the monthly interest rate of this account.
If Sandy first invests \$2000 when she opens the account in September 2013, find the balance at the end of the first month.
Find the value of her savings account at the end of her second month.
Write a recursive rule for V_n in terms of V_{n - 1} that gives the value of the account after n months and an initial investment V_0.
Determine total value of her savings account at the start of September, 2019.