Answer :
To find the explicit formula for the nth term of the sequence defined recursively by [tex]\( a_n = 4a_{n-1} \)[/tex] with the first term given as [tex]\( a_1 = 0.5 \)[/tex], let’s proceed step by step:
1. Understand the Recursive Definition:
The sequence is defined such that each term is 4 times the previous term:
[tex]\[ a_n = 4a_{n-1} \][/tex]
2. Determine the Pattern:
Let's calculate the first few terms to observe a pattern:
- The first term is [tex]\( a_1 = 0.5 \)[/tex]
- The second term is [tex]\( a_2 = 4a_1 = 4 \cdot 0.5 = 2 \)[/tex]
- The third term is [tex]\( a_3 = 4a_2 = 4 \cdot 2 = 8 \)[/tex]
- The fourth term is [tex]\( a_4 = 4a_3 = 4 \cdot 8 = 32 \)[/tex]
3. Identify the General Form:
We see that the terms of the sequence are increasing by a factor of 4 each time. To express this relationship generally, we can write:
[tex]\[ a_2 = 4 \cdot 0.5 = 0.5 \cdot 4^1 \][/tex]
[tex]\[ a_3 = 4 \cdot a_2 = 4 \cdot (0.5 \cdot 4^1) = 0.5 \cdot 4^2 \][/tex]
[tex]\[ a_4 = 4 \cdot a_3 = 4 \cdot (0.5 \cdot 4^2) = 0.5 \cdot 4^3 \][/tex]
From this pattern, we can infer that:
[tex]\[ a_n = 0.5 \cdot 4^{n-1} \][/tex]
4. Confirm the Explicit Formula:
The general formula for the nth term of the sequence is:
[tex]\[ a_n = 0.5 \cdot 4^{n-1} \][/tex]
This matches the option E which is [tex]\( a_n = 0.5 \cdot 4^{n-1} \)[/tex].
Therefore, the correct explicit formula for the nth term of the sequence is given by option E:
[tex]\[ a_n = 0.5 \cdot 4^{(n-1)} \][/tex]
1. Understand the Recursive Definition:
The sequence is defined such that each term is 4 times the previous term:
[tex]\[ a_n = 4a_{n-1} \][/tex]
2. Determine the Pattern:
Let's calculate the first few terms to observe a pattern:
- The first term is [tex]\( a_1 = 0.5 \)[/tex]
- The second term is [tex]\( a_2 = 4a_1 = 4 \cdot 0.5 = 2 \)[/tex]
- The third term is [tex]\( a_3 = 4a_2 = 4 \cdot 2 = 8 \)[/tex]
- The fourth term is [tex]\( a_4 = 4a_3 = 4 \cdot 8 = 32 \)[/tex]
3. Identify the General Form:
We see that the terms of the sequence are increasing by a factor of 4 each time. To express this relationship generally, we can write:
[tex]\[ a_2 = 4 \cdot 0.5 = 0.5 \cdot 4^1 \][/tex]
[tex]\[ a_3 = 4 \cdot a_2 = 4 \cdot (0.5 \cdot 4^1) = 0.5 \cdot 4^2 \][/tex]
[tex]\[ a_4 = 4 \cdot a_3 = 4 \cdot (0.5 \cdot 4^2) = 0.5 \cdot 4^3 \][/tex]
From this pattern, we can infer that:
[tex]\[ a_n = 0.5 \cdot 4^{n-1} \][/tex]
4. Confirm the Explicit Formula:
The general formula for the nth term of the sequence is:
[tex]\[ a_n = 0.5 \cdot 4^{n-1} \][/tex]
This matches the option E which is [tex]\( a_n = 0.5 \cdot 4^{n-1} \)[/tex].
Therefore, the correct explicit formula for the nth term of the sequence is given by option E:
[tex]\[ a_n = 0.5 \cdot 4^{(n-1)} \][/tex]
