Question

The 5th term of a sequence is 2. Each term after the first is half of the previous term. Write an explicit function that models the sequence an.

Answer 1

The explicit function that models the given sequence is
`\(a_n = 2 \cdot \left((1)/(2)\right)^(n-1)\)`.

Step-by-step explanation:

The sequence is defined by the recursive rule where each term is half of the previous term. The 5th term
`(\(a_5\))` is given as 2. To derive the explicit function, we need to find a pattern in the sequence. Let's analyze the terms:

• 
  `\(a_1 = 2\)` (given)

• 
  `\(a_2 = (1)/(2) \cdot a_1 = (1)/(2) \cdot 2 = 1\)`

• 
  `\(a_3 = (1)/(2) \cdot a_2 = (1)/(2) \cdot 1 = (1)/(2)\)`

• 
  `\(a_4 = (1)/(2) \cdot a_3 = (1)/(2) \cdot (1)/(2) = (1)/(4)\)`

• 
  `\(a_5 = (1)/(2) \cdot a_4 = (1)/(2) \cdot (1)/(4) = (1)/(8)\)`

We observe a pattern: each term is obtained by multiplying the previous term by
`\((1)/(2)\)`. Therefore, we can express the nth term
`(\(a_n\)) as \(2 \cdot \left((1)/(2)\right)^(n-1)\)`, where n represents the position of the term in the sequence.

This formula is derived from the fact that each term is obtained by halving the previous term. Starting with
`\(a_1 = 2\)`, the formula correctly generates each subsequent term in the sequence. Hence,
`\(a_n = 2 \cdot \left((1)/(2)\right)^(n-1)\)` is the explicit function that models the given sequence.