Answer to Question #161962 in Discrete Mathematics for mahikah

Question #161962

Write the formula of the following sequence, identify if explicit or recursive formula or both.

1. 1.3,5.7

2. 1, -1, 1, -1

3. 1, 4, 7, 10, 13, 16

4. 0, 3, 8, 15, 24, 25

5. 0, 2, 0, 2, 0, 2 ...

6.1, 1/2, 1/4, 1/8, 1/16

7. 0, 1, 1, 2, 3, 5, 8, 13 ... (fibonacci sequence)


1
Expert's answer
2021-02-12T18:08:26-0500

Let us write the explicit and recursive formulas of the following sequences:


1. 1.3, 5.7,...

explicit formula: "a_n=-3.1+4.4n", recursive formula: "a_{n+1}=a_n+4.4, \\ a_1=1.3"


2. 1, -1, 1, -1,...

explicit formula: "a_n=(-1)^{n-1}", recursive formula: "a_{n+1}=-a_n, \\ a_1=1"


3. 1, 4, 7, 10, 13, 16,...

explicit formula: "a_n=-2+3n", recursive formula: "a_{n+1}=a_n+3, \\ a_1=1"


4. 0, 3, 8, 15, 24, ...

explicit formula: "a_n=n^2-1", recursive formula: "a_{n+1}=a_n+2n-1, \\ a_1=0"


5. 0, 2, 0, 2, 0, 2, ...

explicit formula: "a_n=1+(-1)^n", recursive formula: "a_{n+1}=a_n+2(-1)^{n+1}, \\ a_1=0"


6. 1, 1/2, 1/4, 1/8, 1/16,...

explicit formula: "a_n=2^{1-n}", recursive formula: "a_{n+1}=\\frac{a_n}{2}, \\ a_1=1"


7. 0, 1, 1, 2, 3, 5, 8, 13, ...

explicit formula: "a_n=\\frac{(1+\\sqrt{5})^{n-1}-(1-\\sqrt{5})^{n-1}}{2^{n-1}\\sqrt{5}}", recursive formula: "a_{n+2}=a_{n+1}+a_{n}, \\ a_1=0, a_2=1"




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