Question

Dada a P.A. (5, 2, -1,...) determine:
a] o 31º termo
b] o termo geral
c] a soma dos 10 primeiros termos
d] a soma dos N primeiros termos

Cálculo, por favor

Answer 1

$P.A~(5,~2,-1,...)$

$a_{1}=5\\a_{2}=2\\\\r=a_{2}-a_{1}=2-5=-3$

a)

$a_{n}=a_{1}+(n-1)r\\a_{31}=a_{1}+(31-1)r\\a_{31}=a_{1}+30r\\a_{31}=5+30*(-3)\\a_{31}=5-90\\a_{31}=-85$

b)

$a_{n}=a_{1}+(n-1)r\\a_{n}=5+(n-1)*(-3)\\a_{n}=5-3n+3\\a_{n}=8-3n$

c)

$a_{10}=8-3*10=8-30=-22\\\\S_{n}=(a_{1}+a_{n})*n/2\\S_{10}=(a_{1}+a_{10})*10/2\\S_{10}=(5-22)*5\\S_{10}=(-17)*5\\S_{10}=-85$

d)

$S_{n}=(a_{1}+a_{n})*n/2\\S_{n}=(5+8-3n)*n/2\\S_{n}=(13-3n)*n/2\\S_{n}=(13n-3n^{2})/2$