Answer :
Let's analyze the given explicit formula for the sequence:
[tex]\[ f(n) = 70(-2)^{n-1} \][/tex]
We want to convert this explicit formula into a recursive formula, which means expressing [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n-1) \)[/tex].
First, let's observe the pattern:
- The first term [tex]\( f(1) \)[/tex] when [tex]\( n = 1 \)[/tex] is:
[tex]\[ f(1) = 70(-2)^{1-1} = 70(-2)^0 = 70 \cdot 1 = 70 \][/tex]
- The second term [tex]\( f(2) \)[/tex] when [tex]\( n = 2 \)[/tex] is:
[tex]\[ f(2) = 70(-2)^{2-1} = 70(-2)^1 = 70 \cdot (-2) = -140 \][/tex]
- The third term [tex]\( f(3) \)[/tex] when [tex]\( n = 3 \)[/tex] is:
[tex]\[ f(3) = 70(-2)^{3-1} = 70(-2)^2 = 70 \cdot 4 = 280 \][/tex]
- The fourth term [tex]\( f(4) \)[/tex] when [tex]\( n = 4 \)[/tex] is:
[tex]\[ f(4) = 70(-2)^{4-1} = 70(-2)^3 = 70 \cdot (-8) = -560 \][/tex]
Now, let's analyze the relationship between consecutive terms. We notice that:
[tex]\[ \begin{aligned} f(2) &= -2 \cdot f(1) & = -2 \cdot 70 & = -140 \\ f(3) &= -2 \cdot f(2) & = -2 \cdot (-140) & = 280 \\ f(4) &= -2 \cdot f(3) & = -2 \cdot 280 & = -560 \end{aligned} \][/tex]
Therefore, we can see a clear pattern where each term is obtained by multiplying the previous term by -2.
Thus, the recursive formula for [tex]\( n > 1 \)[/tex] is:
[tex]\[ \begin{aligned} f(1) &= 70 \\ f(n) &= -2 \cdot f(n-1) \quad \text{for} \quad n > 1 \end{aligned} \][/tex]
From the given choices, the correct recursive formula is:
[tex]\[ f(n) = -2 \cdot f(n-1) \quad \text{for} \quad n > 1 \][/tex]
Hence, the correct answer is:
[tex]\[ f(n) = -2 f(n-1) \][/tex]
[tex]\[ f(n) = 70(-2)^{n-1} \][/tex]
We want to convert this explicit formula into a recursive formula, which means expressing [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n-1) \)[/tex].
First, let's observe the pattern:
- The first term [tex]\( f(1) \)[/tex] when [tex]\( n = 1 \)[/tex] is:
[tex]\[ f(1) = 70(-2)^{1-1} = 70(-2)^0 = 70 \cdot 1 = 70 \][/tex]
- The second term [tex]\( f(2) \)[/tex] when [tex]\( n = 2 \)[/tex] is:
[tex]\[ f(2) = 70(-2)^{2-1} = 70(-2)^1 = 70 \cdot (-2) = -140 \][/tex]
- The third term [tex]\( f(3) \)[/tex] when [tex]\( n = 3 \)[/tex] is:
[tex]\[ f(3) = 70(-2)^{3-1} = 70(-2)^2 = 70 \cdot 4 = 280 \][/tex]
- The fourth term [tex]\( f(4) \)[/tex] when [tex]\( n = 4 \)[/tex] is:
[tex]\[ f(4) = 70(-2)^{4-1} = 70(-2)^3 = 70 \cdot (-8) = -560 \][/tex]
Now, let's analyze the relationship between consecutive terms. We notice that:
[tex]\[ \begin{aligned} f(2) &= -2 \cdot f(1) & = -2 \cdot 70 & = -140 \\ f(3) &= -2 \cdot f(2) & = -2 \cdot (-140) & = 280 \\ f(4) &= -2 \cdot f(3) & = -2 \cdot 280 & = -560 \end{aligned} \][/tex]
Therefore, we can see a clear pattern where each term is obtained by multiplying the previous term by -2.
Thus, the recursive formula for [tex]\( n > 1 \)[/tex] is:
[tex]\[ \begin{aligned} f(1) &= 70 \\ f(n) &= -2 \cdot f(n-1) \quad \text{for} \quad n > 1 \end{aligned} \][/tex]
From the given choices, the correct recursive formula is:
[tex]\[ f(n) = -2 \cdot f(n-1) \quad \text{for} \quad n > 1 \][/tex]
Hence, the correct answer is:
[tex]\[ f(n) = -2 f(n-1) \][/tex]
