Question

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)

Answer 1

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: $a_{n+2}=a_{n+1}+a_{n}, \ a_1=0, a_2=1$