What expression can be used to find the fifteenth term of the sequence 2, 6, 18, 54?

To derive the fifteenth term of the sequence 2, 6, 18, 54, we first need to identify the pattern within the sequence. Observing the numbers:

• 2 is the first term (T₁)
• 6 is the second term (T₂)
• 18 is the third term (T₃)
• 54 is the fourth term (T₄)

Now, let’s look at how each term is derived from its predecessor. This appears to be a geometric sequence where each term is multiplied by 3 to get the next term:

• T₁ = 2
• T₂ = T₁ × 3 = 2 × 3 = 6
• T₃ = T₂ × 3 = 6 × 3 = 18
• T₄ = T₃ × 3 = 18 × 3 = 54

From this, we can derive a general formula for the nth term of the sequence. Each term can be expressed as:

T_n = T₁ × 3^(n-1)

Substituting what we know:

T_n = 2 × 3^(n-1)

To find the fifteenth term (n = 15):

T₁₅ = 2 × 3^(15-1) = 2 × 3¹⁴

Calculating 3¹⁴, we find that:

3¹⁴ = 4782969

Therefore:

T₁₅ = 2 × 4782969 = 9565938

In conclusion, the expression to find the fifteenth term of this sequence is:

T = 2 × 3¹⁴ = 9565938