Question

Encontre a posição dos termos indicados em cada sequência:

a- (2, 7, 12, ..., 82, ...)

b- (-3, -1, ..., 67, ...)

c- (100, 96, 92, ..., -32)

d- (32, 39, ..., 200,...)

e- (7, 2, ..., -63, ...)

​

Answer 1

Considerando que todas sejam PA's, temos:

a)

$razao=a_2-a_1\\\\razao=7-2\\\\\boxed{razao=5}\\\\\\Utilizando~a~equacao~do~termo~geral:\\a_n=a_1+(n-1).r\\\\82=2+(n-1).5\\\\82=2+5n-5\\\\5n=82-2+5\\\\n=\frac{85}{5}\\\\\boxed{n=17^o~termo}$

b)

$razao=a_2-a_1\\\\razao=-1-(-3)\\\\\boxed{razao=2}\\\\\\Utilizando~a~equacao~do~termo~geral:\\a_n=a_1+(n-1).r\\\\67=2+(n-1).2\\\\67=-3+2n-2\\\\2n=67+3+2\\\\n=\frac{72}{2}\\\\\boxed{n=36^o~termo}$

c)

$razao=a_2-a_1\\\\razao=96-100\\\\\boxed{razao=-4}\\\\\\Utilizando~a~equacao~do~termo~geral:\\a_n=a_1+(n-1).r\\\\-32=100+(n-1).(-4)\\\\-32=100-4n+4\\\\-4n=-32-100-4\\\\n=\frac{-136}{-4}\\\\\boxed{n=34^o~termo}$

d)

$razao=a_2-a_1\\\\razao=39-32\\\\\boxed{razao=7}\\\\\\Utilizando~a~equacao~do~termo~geral:\\a_n=a_1+(n-1).r\\\\200=32+(n-1).7\\\\200=32+7n-7\\\\7n=200-32+7\\\\n=\frac{175}{7}\\\\\boxed{n=25^o~termo}$

e)

$razao=a_2-a_1\\\\razao=2-7\\\\\boxed{razao=-5}\\\\\\Utilizando~a~equacao~do~termo~geral:\\a_n=a_1+(n-1).r\\\\-63=7+(n-1).(-5)\\\\-63=7-5n+5\\\\-5n=-63-7-5\\\\n=\frac{-75}{-5}\\\\\boxed{n=15^o~termo}$