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1.)Given the recursive definition LaTeX: a_{n+1}=2\cdot a_n+1a n + 1 = 2 ⋅ a n + 1, and that LaTeX: a_1=1a 1 = 1, find LaTeX: a_{10}a 10


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2.)Given the explicit definition: LaTeX: t_n=n^2-2nt n = n 2 − 2 n, find LaTeX: t_4

Respuesta :

Answer:

1) 1023

2) 8

Step-by-step explanation:

1)

Given:

[tex]a_{n+1}=2\cdot a_n + 1[/tex] and [tex]a_1 = 1[/tex]

Then,  the next table can be computed (only the first terms are explicitely shown)

n      [tex]a_n[/tex]

1       1

2      [tex]a_{2}=2\cdot 1 + 1 =[/tex] 3

3      [tex]a_{3}=2\cdot 3 + 1 =[/tex]7

4      15

5      31

6      63

7      127

8      255

9      511

10     1023

2)

Given

[tex]t_n=n^2-2n[/tex]

Then,  the next table can be computed

n      [tex]t_n[/tex]

1       [tex]t_1=1^2-2(1) = [/tex]-1

2      [tex]t_2=2^2-2(2) = [/tex]0

3      [tex]t_3=3^2-2(3) = [/tex]3

4      [tex]t_4=4^2-2(4) = [/tex]8

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