Question

determine o número de termos da p.a ( - 6, - 9, - 12,..,-69)

Answer 1

Na P.A. $\left(-6,\,-9,\,-12,\,\ldots,\,-69 \right )$, temos

$a_{1}=-6\\ \\ a_{n}=-69\\ \\ r=-9-\left(-6\right) \Rightarrow r=-3\\ \\ \\ a_{n}=a_{1}+\left(n-1 \right ) \cdot r\\ \\ -69=-6+\left(n-1 \right )\cdot\left(-3 \right )\\ \\ -3\cdot \left(n-1\right)=-69+6\\ \\ -3\cdot \left(n-1\right)=-63\\ \\ n-1=\dfrac{-63}{-3}\\ \\ n-1=21\\ \\ n=21+1\\ \\ \boxed{n=22 \text{ termos}}$

Answer 2

An = a1 + (n-1).r
razão r = a2 - a1
r = -9 - (-6)
r = -9 + 6
r = -3
-69 = -6 - 3(n - 1)
-69 = -6 - 3n + 3
-69 = -3 - 3n
-3n - 3 = - 69
-3n = -69 + 3
-3n = -66
   n = 66 / 3
   n = 22
Resp: 22 termos