Match the sequence and recursive expression to its explicit expression. (2 points)
f(n) = ール
f (n) = 2n - 3
f (n) = 2n + 2
J(n)=-7+5
# {4, 6, 8, 10, ... ] where / (1) = 4 and f (n) = f (n - 1) + 2 torn > 1
= {4, 3, 2, 1, . . . ] where f (1) - 4 and f(n) = f (n - 1) - 1 for n. > 1
= {-1, 1, 3, 5, . . . } where f (1) = -1 and f(n) = f (n - 1) + 2 for n. > 1
1-1, -2, -3, -4, .7where f(1) - -1 and f(n) = ナ(九 - 1) - 1 for
The correct matches are:
1. {4, 6, 8, 10, ... ] where f(n) = f(n-1) + 2 for n > 1 matches f(n) = 2n + 2
2. {4, 3, 2, 1, ... ] where f(1) = 4 and f(n) = f(n-1) - 1 for n > 1 matches f(n) = 5 - n
3. {-1, 1, 3, 5, ... } where f(1) = -1 and f(n) = f(n-1) + 2 for n > 1 matches f(n) = 2n - 3
4. 1, -2, -3, -4, ... where f(1) = -1 and f(n) = f(n-1) - 1 for n > 1 matches f(n) = -n+1
The explicit expressions for the given sequences are as follows:
1. Sequence: {4, 6, 8, 10, ... }
Recursive Expression: f(n) = f(n-1) + 2 (where f(1) = 4)
Explicit Expression: f(n) = 2n + 2
2. Sequence: {4, 3, 2, 1, ... }
Recursive Expression: f(n) = f(n-1) - 1 (where f(1) = 4)
Explicit Expression: f(n) = 5 - n
3. Sequence: {-1, 1, 3, 5, ... }
Recursive Expression: f(n) = f(n-1) + 2 (where f(1) = -1)
Explicit Expression: f(n) = 2n - 3
4. Sequence: {-1, -2, -3, -4, ... }
Recursive Expression: f(n) = f(n-1) - 1 (where f(1) = -1)
Explicit Expression: f(n) = -n - 2
To match the sequences and recursive expressions to their explicit expressions, we can analyze each sequence and find the pattern.
Let's start with the first sequence: {4, 6, 8, 10, ...}
The recursive expression given is f(n) = f(n - 1) + 2, where f(1) = 4. This means the next term is obtained by adding 2 to the previous term.
By applying this recursive rule, we can see that the explicit expression for this sequence is f(n) = 2n + 2. Thus, the correct match is:
Sequence: {4, 6, 8, 10, ...}
Recursive Expression: f(n) = f(n - 1) + 2
Explicit Expression: f(n) = 2n + 2
Moving on to the second sequence: {4, 3, 2, 1, ...}
The recursive expression given is f(n) = f(n - 1) - 1, where f(1) = 4. This means the next term is obtained by subtracting 1 from the previous term.
By applying this recursive rule, we can see that the explicit expression for this sequence is f(n) = 5 - n. Thus, the correct match is:
Sequence: {4, 3, 2, 1, ...}
Recursive Expression: f(n) = f(n - 1) - 1
Explicit Expression: f(n) = 5 - n
Lastly, let's consider the third sequence: {-1, 1, 3, 5, ...}
The recursive expression given is f(n) = f(n - 1) + 2, where f(1) = -1. This means the next term is obtained by adding 2 to the previous term.
By applying this recursive rule, we can see that the explicit expression for this sequence is f(n) = 2n - 3. Thus, the correct match is:
Sequence: {-1, 1, 3, 5, ...}
Recursive Expression: f(n) = f(n - 1) + 2
Explicit Expression: f(n) = 2n - 3
Therefore, the matches are:
1) Sequence: {4, 6, 8, 10, ...}, Recursive Expression: f(n) = f(n - 1) + 2, Explicit Expression: f(n) = 2n + 2
2) Sequence: {4, 3, 2, 1, ...}, Recursive Expression: f(n) = f(n - 1) - 1, Explicit Expression: f(n) = 5 - n
3) Sequence: {-1, 1, 3, 5, ...}, Recursive Expression: f(n) = f(n - 1) + 2, Explicit Expression: f(n) = 2n - 3