Math Problem Statement

Find the explicit formula for 1,6,36,216...

Solution

Let's analyze the sequence 1,6,36,216,1, 6, 36, 216, \dots to determine its explicit formula:

Step 1: Recognize the pattern

  • Start with the sequence: 1,6,36,216,1, 6, 36, 216, \dots
  • Notice that each term seems to grow quickly. To confirm, let's check the ratio between consecutive terms.

a2a1=61=6,a3a2=366=6,a4a3=21636=6\frac{a_2}{a_1} = \frac{6}{1} = 6, \quad \frac{a_3}{a_2} = \frac{36}{6} = 6, \quad \frac{a_4}{a_3} = \frac{216}{36} = 6

Since the ratio is constant (r=6r = 6), the sequence is a geometric progression.

Step 2: General formula for geometric sequences

The general formula for a geometric sequence is:

an=a1rn1a_n = a_1 \cdot r^{n-1}

  • Here:
    • a1=1a_1 = 1 (the first term),
    • r=6r = 6 (common ratio).

Substitute into the formula:

an=16n1or simplyan=6n1a_n = 1 \cdot 6^{n-1} \quad \text{or simply} \quad a_n = 6^{n-1}

Step 3: Match with the options

Looking at the options:

  • Option D matches an=6n1a_n = 6^{n-1}.

Final Answer:

Option D is correct.


Do you want further clarification or a detailed breakdown of any step?

Here are five related questions you may want to explore:

  1. What is the general formula for a geometric sequence?
  2. How can you identify a geometric sequence from given terms?
  3. How do you find the sum of the first nn-terms of a geometric sequence?
  4. What is the difference between arithmetic and geometric sequences?
  5. How do you determine the common ratio of a geometric sequence?

Tip: In geometric sequences, the ratio between consecutive terms remains constant, which simplifies finding the explicit formula.

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Math Problem Analysis

Mathematical Concepts

Geometric Sequences
Explicit Formula
Common Ratio

Formulas

a_n = a_1 * r^(n-1)

Theorems

General formula for geometric sequences

Suitable Grade Level

Grades 9-11