10.03 Geometric sequences
Write down the next two terms for the following sequences:
4, 12, 36, \ldots
12, - 48, 192, \ldots
- 1, 8, - 64, \ldots
- 6, 9, -\dfrac{27}{2},\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.
Explain how the common ratio of a geometric sequence can be found.
Suppose u_{1}, u_{2}, u_{3}, u_{4}, u_{5},\ldots is a geometric sequence.
Is u_{1}, u_{3}, u_{5},\ldots also a geometric sequence? Explain your answer.
Consider the first four terms of the following geometric sequences:
Evaluate \dfrac{u_2}{u_1}.
Evaluate \dfrac{u_3}{u_2}.
Evaluate \dfrac{u_4}{u_3}.
Hence, find u_5.
- 4 , - 8 , - 16 , - 32 , \ldots
2, - 6 , 18, - 54 , \ldots
- 64 , - 16 , - 4 , -1, \ldots
State the common ratio between the terms of the following sequences:
9, 36, 144, 576, \ldots
- 6 , - 42 , - 294 , - 2058 , \ldots
2, - 16, 128, - 1024, \ldots
- 70.4 , - 17.6 , - 4.4 , - 1.1 ,\ldots
Write the first 5 terms of the following sequences given the first term and the common ratio:
First term: - 2, common ratio: 3
First term: 1.3, common ratio: - 4.
First term: 700\,000, common ratio: 1.04.
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.
The first term is 700\,000 and the common ratio is 1.04.
The first term is - 2 and the common ratio is 3.
The first term is 1.3 and the common ratio is - 4.
Find the missing terms in the following geometric progressions:
- 5, \, x, \, - 80, \, 320, \, y
a, \, b, \, \dfrac{3}{25}, \, - \dfrac{3}{125}, \, c
For each of the following pairs of terms in a geometric progression:
Find the possible values of r.
Find the value of u_1.
Find the general rule for u_n, for r \gt 0.
u_3 = 18 and u_5 = 162
u_4 = 32 and u_6 = 128
Suppose you save \$1 on the first day of a month, \$2 on the second day, \$4 on the third day, \$8 on 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 6th day of the month?
How much will you put aside for savings on the 10th day of the month?
The average daily growth of a seedling is 10\% per day. A seedling measuring 6 \text{ cm} in height is planted.
Determine the height of the seedling at the end of Day 1.
Find the height of the seedling 2 days after it is planted.
Write a recursive rule for H_n, defining the height of the seedling n days after it is planted, and an initial condition H_0.