Question

A soma dos 20 primeiros termos de uma pa finita e 710 .Se o 1° termo dessa parte e a1 = 7, calcule o 10° termo.

Answer 1

Temos que:

$a_{n} = a_{1} + (n-1)r \\ \\ a_{20} = 7 + 19r$

Assim:

$S_{20} = (a_{1} + a_{20})* \frac{20}{2} \\ \\ 710= (7 + a_{20})*10 \\ \\ 71 = 7 +a_{20} \\ \\ a_{20} = 64 \\ \\ 7 + 19r = 64 \\ \\ r = \frac{57}{19} \\ \\ r = 3$

Logo:

$a_{10} = a_{1} + 9r \\ \\ a_{10} = 7 + 9*3 \\ \\ a_{10} = 7 + 27 \\ \\ a_{10} = 34$

Answer 2

$Sabemos \ que: \\\\ n=20 \ | \ S_n=710 \ | \ a_1=7 \ | \ a_{10}= \ ? \\\\ S_n= \frac{(a_1+a_n)n}{2} \\ a_n=a_k+(n-k)r \\\\ Achando \ o \ valor \ de \ a_{20} \ (a_{n}): \\\\ 710= \frac{(7+a_{20})20}{2} \\ 710=(7+a_{20})10 \\ 7+a_{20}= \frac{710}{10} \\ a_{20}=71-7 \\ a_{20}=64 \\\\ Achando \ o \ valor \ de \ r: \\\\ a_{20}=a_1+(20-1)r \\ 64=7+19r \\ 19r=64-7 \\ r= \frac{57}{19} \ --\ \textgreater \ \ r=3 \\\\ Achando \ o \ valor \ de \ a_{10}: \\\\ a_{10}=a_1+(10-1)r \\ a_{10}=7+(9)3 \\ a_{10}=7+27 \\ a_{10}=34$