Question

dado um P.A com o 5 termo valendo 30 e o 20 termo valendo 50 quanto vale o 9 termo dessa progressão

Answer 1

encontrar a razão da PA

an = ak + ( n - k ).r
30 = 50 + ( 5 - 20 ) . r
30 = 50 - 15.r
30 - 50 = -15. r
-20 / -15 = r
r = 20/15
r = 4/3

====

an = a1 + ( n - 1 ) . r
30 = a1 + ( 5 - 1 ) . 4/3
30 = a1 + 4 . 4/3
30 = a1 + 16/3
30 - 16/3 = a1
a1 = 74/3

===

Termo geral:

an = a1 + ( n -1) . r
an = 74/3 +  ( n -1) . 4/3
an = 74/3 + (4n - 4) / 3
an = (4n + 70 / 3)

===
Encontrar o valor do termo a9: (Nono termo).

an = (4n + 70 / 3)
a9 = (4.9 + 70) / 3
a9 = (36 + 70) / 3
a9 = 106/3

Answer 2

Dados:

๏ $\mathsf{a_{5}=30}$
๏ $\mathsf{a_{20}=50}$

Inicialmente iremos determinar o valor da razão, utilizando a formula geral de uma P.A:


$\mathsf{a_n=a_k+\left(n-k\right)r}\\\mathsf{a_{20}=a_5+\left(20-5\right)r}\\\mathsf{50=30+15r}\\\mathsf{15r=50-30}\\\mathsf{15r=20}\\\\\mathsf{r=\dfrac{20}{15}}\\\\\mathsf{r=\dfrac{4}{3}}$

Agora aplicando na formula para determinar o valor do $\mathsf{a_9}$:

$\mathsf{a_9=a_5+\left(9-5\right)\cdot \dfrac{4}{3}}\\\\\\\mathsf{a_9=30+\left(4\cdot \dfrac{4}{3}\right)}\\\\\\\mathsf{a_9=30+\dfrac{16}{3}}\\\\\\\mathsf{a_9=\dfrac{30\cdot 3}{3}+\dfrac{16}{3}}\\\\\\\mathsf{a_9=\dfrac{90}{3}+\dfrac{16}{3}}\\\\\\\mathsf{a_9=\dfrac{90+16}{3}}\\\\\\\boxed{\boxed{\mathsf{a_9=\dfrac{106}{3}}}}$