Math Problem Statement
geometric sequences
Solution
Let's analyze and solve the questions related to the image of geometric sequences.
Q1:
The given sequence is . We need to find the next three terms.
- Step 1: Find the common ratio of the sequence.
- To find the common ratio, divide the second term by the first term:
- Step 2: Apply the common ratio to find the next terms:
- Fourth term:
- Fifth term:
- Sixth term:
Thus, the next three terms are 0.4, 0.08, and 0.016.
Q2:
-
What is the common ratio ?
- From the previous calculation, the common ratio .
-
What is the formula for the -th term? The general formula for the -th term of a geometric sequence is: For this sequence:
Q3: Work out the first three terms of the sequences with the given formulas.
-
For the sequence :
- First term:
- Second term:
- Third term:
-
For the sequence :
- First term:
- Second term:
- Third term:
Final Question:
- The second term of a geometric sequence is 10.5 and the third term is 73.5.
- We know that and . Using the formula for the -th term: Divide by to find the common ratio: So, . Now use this to find :
Thus, the first term is 1.5 and the common ratio is 7.
Would you like more details or have any questions on this? Here are some additional related questions:
- How do you derive the formula for the -th term of a geometric sequence?
- Can you find the sum of the first 10 terms of the sequence ?
- How does the common ratio affect the behavior of a geometric sequence (e.g., increasing, decreasing)?
- Can you explore geometric sequences with negative common ratios?
- What happens if the common ratio is greater than 1 or less than 1?
Tip: In geometric sequences, if , the terms approach 0 as increases!
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Math Problem Analysis
Mathematical Concepts
Geometric Sequences
Ratios
Nth Term Formula
Formulas
Common Ratio: r = T2 / T1
Nth term formula: Tn = T1 * r^(n-1)
Theorems
Geometric Sequence Theorem
Suitable Grade Level
Grades 9-11