Question

Lat \(a_{n}\) be the nth term of a geometric sequence. If a7=32 and a9=8, which of the following must be true?
A. a1>0
B. a1-a2>0
C. a2+a3+a4+...+a100 > 0

@RadEn @ganeshie8

Answer 1

a_n = a1r^(n-1)

a7 = 32 ----> a1r^6 = 32 ... (1)
a9 = 8 ----> a1r^8 = 8 ... (2)

a1r^8/a1r^6 = 8/32
r^2 = 1/4
r = +-1/2 (r = 1/2 or r = -1/2)

now, can you find the value of a1 ?

Answer 2

@kryton1212

Answer 3

logic : when we divide positive numbers, we get a positive ratio, so, r>0
hence, every term of the series will be positive, so
A and C must be true.
Math : shown by Redn

for B, since a9<a7, all higher terms will be less than lower terms, like \(a_{n+1}<a_n\)
so, a1>a2

Answer 4

wait a moment, i am thinking what did @RadEn has said

Answer 5

if we subtitute the possible value of r ( + and -) to one of equation above (1 or 2), i sure there are 2 possible values of a1, are a1>0 or a1< 0
@hartnn

Answer 6

so, i think A not always be true

Answer 7

clear. Why B is true then?

Answer 8

i didnt say that --"

Answer 9

but i think B is true.
HAHAHA

Answer 10

looks if r > 0 then a1 a>0 too while
if r < 0 then a1 < 0 too

Answer 11

what does it mean?

Answer 12

wait, looks hartnn is right :)
a1r^6 = 32
if we subtitute r=1/2 or r=-1/2, r^6 must be possitive, thus
a1 * (positive number) = 32
a1 must be positive too (in other words a1 > 0)
so, A is correct

Answer 13

lol ..

Answer 14

haha, sorry

Answer 15

how about B ?

Answer 16

hmmm...
a1r^6 = 32
a1 (1/2)^6 = 32
a1 = 32/(1/2)^6 = 32 * 2^6 = 2048

now, a2 = a1r
if we take r = 1/2, obviously a2 be (+), is
a2 = a1r = 2048 * 1/2 = 1024
if r = -1/2 then a2 be negative :
a2 = a1r = 2048 * (-1/2) = -1024

the 1st possible of a1 - a2 are
a1 - a2 = 2048 - 1024 = 1024 (it is positive)

the 2nd possible :
a1-a2 = 2048 - (-1024) = 3070 (positive too)

so, B is correct too :)

Answer 17

and like hartnn said above that a_(n+1) < a_n, it means the sum of
a2+a3+a4+...+a100 > 0 be true

so, A,B, and C all correct

Answer 18

Looks like logic is hard to understand initially...

Answer 19

but looks we were make mistake again, @hartnn :)
for r = 1/2. obviously all terms a1,a2,a3,a4,... and so on be positive
and obviously the sum of a2+a3+a4+...+a100 > 0

but if r = -1/2, we get all terms respectively :
a1 = 2048
a2 = -1024
a3 = 512
a4 = -128
a5 = 64
a6 = -32
....
so on

the option C told us the sum started from a2 (without term a1), so its sum :
a2 + a3 + a4 + a5 + a6 + .... + a100 = -1024 + 523 - 128 + 64 - 32 + ... - ?
i sure the sum above become negative value (< 0) not positive value (>0)

sorry again, @kryton1212 . im not careful this season
so, A and B are correct but not always for C