Math Problem Statement
solve the exponential sequence 2,4,8,16,32,64
Solution
The given sequence is:
This is an exponential sequence where each term is multiplied by a constant ratio to get the next term. Let's analyze it step-by-step.
1. Identifying the pattern:
Each term is obtained by multiplying the previous term by .
This is a geometric progression with a common ratio .
2. General formula for a geometric sequence:
The -th term of a geometric sequence is given by: where:
- = first term = 2
- = common ratio = 2
3. Formula for this sequence:
4. Verifying:
- ,
- ,
- ,
- ,
- ,
- ,
The formula works perfectly!
5. How to find any term:
If you want to find any specific term in the sequence, just plug in the value of . For example, the 10th term would be:
Would you like to explore this further or ask anything else?
Related Questions:
- What is the sum of the first terms in a geometric sequence?
- How can we determine the sum of an infinite geometric series?
- How does a geometric sequence differ from an arithmetic sequence?
- What are some real-world applications of geometric sequences?
- Can a geometric sequence have a negative common ratio?
Tip: Exponential growth increases rapidly—be careful with calculations involving large exponents to avoid errors!
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Geometric Progression
Sequences
Formulas
Geometric sequence formula: a_n = a_1 * r^(n - 1)
Exponential formula for the sequence: a_n = 2^n
Theorems
None directly involved
Suitable Grade Level
Grades 6-8
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