Math Problem Statement

solve the exponential sequence 2,4,8,16,32,64

Solution

The given sequence is:
2,4,8,16,32,642, 4, 8, 16, 32, 64

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 22.

42=2,84=2,168=2,and so on...\frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2, \quad \text{and so on...}

This is a geometric progression with a common ratio r=2r = 2.

2. General formula for a geometric sequence:

The nn-th term ana_n of a geometric sequence is given by: an=a1rn1a_n = a_1 \cdot r^{n - 1} where:

  • a1a_1 = first term = 2
  • rr = common ratio = 2

3. Formula for this sequence:

an=22n1=2na_n = 2 \cdot 2^{n - 1} = 2^n

4. Verifying:

  • n=1n = 1, a1=21=2a_1 = 2^1 = 2
  • n=2n = 2, a2=22=4a_2 = 2^2 = 4
  • n=3n = 3, a3=23=8a_3 = 2^3 = 8
  • n=4n = 4, a4=24=16a_4 = 2^4 = 16
  • n=5n = 5, a5=25=32a_5 = 2^5 = 32
  • n=6n = 6, a6=26=64a_6 = 2^6 = 64

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 nn. For example, the 10th term a10a_{10} would be: a10=210=1024a_{10} = 2^{10} = 1024

Would you like to explore this further or ask anything else?


Related Questions:

  1. What is the sum of the first nn terms in a geometric sequence?
  2. How can we determine the sum of an infinite geometric series?
  3. How does a geometric sequence differ from an arithmetic sequence?
  4. What are some real-world applications of geometric sequences?
  5. 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