1. ## Question about geometric sequence

I want to know how I can solve this question:
IfYnis a geometric sequence such that,y1+y2=0 then the right statement of
the following is:
A) Y5+Y7=0

B)Y6+Y8=0

C)2Y2+3Y1=0
D)Y2+2Y3+Y4=0

2. Hello, Diligent_Learner!

$y_n$ is a geometric sequence such that: . $y_1+y_2\:=\:0$
then which of the following is correct?

$A)\;\;y_5\;+\;y_7 \:=\:0\qquad B)\;\;y_6\;+\;y_8\:=\:0 \qquad C)\;\;2y_2\;+\;3y_1\:=\:0 \qquad D)\;\;y_2\;+\;2y_3\;+\;y_4\:=\:0$

A geometric series has the form: . $a,\:ar,\;ar^2,\;ar^3,\:\hdots\quad \text{ where }a \neq 0$

Since $y_1+y_2 \:=\:0$, we have: . $a + ar \:=\:0 \quad\Rightarrow\quad a(1+r) \:=\:0$

. . Since $a \neq 0$, then $r+1 \:=\:0 \quad\Rightarrow\quad r = \text{-}1$

Hence, the sequence is: . $a,\:\text{-}a,\:a,\:\text{-}a,\:a,\: \hdots$

. . That is: . $y_n \;=\;\bigg\{ \begin{array}{cc}a & \text{for odd }n \\ \text{-}a & \text{for even }n \end{array}$

$A)\;\;y_5+y_7 \:=\:a + a \:=\:2a$ . . . false

$B)\;\;y_6+y_8 \:=\:(\text{-}a) + (\text{-}a) \:=\:-2a$ . . . false

$C)\;\;2y_2 + 3y_1 \:=\:2(\text{-}a) + 3(a) \:=\:a$ . . . false

$D)\;\;y_2 +2y_3 + y_4 \:=\:(\text{-}a) + 2(a) + (\text{-}a) \:=\:0$ . . . True!

3. Originally Posted by Soroban
Hello, Diligent_Learner!

A geometric series has the form: . $a,\:ar,\;ar^2,\;ar^3,\:\hdots\quad \text{ where }a \neq 0$

Since $y_1+y_2 \:=\:0$, we have: . $a + ar \:=\:0 \quad\Rightarrow\quad a(1+r) \:=\:0$

. . Since $a \neq 0$, then $r+1 \:=\:0 \quad\Rightarrow\quad r = \text{-}1$

Hence, the sequence is: . $a,\:\text{-}a,\:a,\:\text{-}a,\:a,\: \hdots$

. . That is: . $y_n \;=\;\bigg\{ \begin{array}{cc}a & \text{for odd }n \\ \text{-}a & \text{for even }n \end{array}$

$A)\;\;y_5+y_7 \:=\:a + a \:=\:2a$ . . . false

$B)\;\;y_6+y_8 \:=\\text{-}a) + (\text{-}a) \:=\:-2a" alt="B)\;\;y_6+y_8 \:=\\text{-}a) + (\text{-}a) \:=\:-2a" /> . . . false

$C)\;\;2y_2 + 3y_1 \:=\:2(\text{-}a) + 3(a) \:=\:a$ . . . false

$D)\;\;y_2 +2y_3 + y_4 \:=\\text{-}a) + 2(a) + (\text{-}a) \:=\:0" alt="D)\;\;y_2 +2y_3 + y_4 \:=\\text{-}a) + 2(a) + (\text{-}a) \:=\:0" /> . . . True!

Thank you very much ..really I understand the question