What is the rule for pattern 1,3,9 ,27 ,84

Question

What's the rule for the pattern 1,3,9,27,84

Answer 1

Assuming the sequence is : 1, 3, 9, 27, 81

Often, we look for the well-known sequences:

(1) Is it an arithmetic sequence (with a common difference)? No

(2) Is it a geometric sequence (with a common ratio)? Yes (if T₅=81)
Common ratio is 3 [each term is 3 times the previous term]

(3) [note: others are defined with expressions or recursively]


Let's express terms as T_n with starting at 1:

T_n = 3^n-1     for n≥1

Or

T₁ = 1
T_n = 3T_n-1      for n>1

Answer 2

Malisa,

 

Let's look at the first 3 terms:

 

1 can be rewritten as 3⁰

 

3 can be rewritten as 3¹

 

9 can be rewritten as 3²

 

So, each term (1st, 2nd, 3rd, etc), can be written as:

 

3^n-1 where n is the place of the term in the sequence.

 

The first term then is 3^1-1 = 3⁰ = 1

 

Make sense?