Question

Let a,b,c be +ve integers. If b=sqrt(ac), which of the following must be true?
A. loga^2, logb^2, logc^2 is an arithmetic sequence.
B. a^3, b^3, c^3 is a geometric sequence.
C. 4^a, 4^b, 4^c is a geometric sequence.

Answer 1

@phi

Answer 2

b= sqrt(ac) means a,b,c are a geometric sequence.

remember yesterday ?
\[ \frac{a}{b} = \frac{b}{c} \\ b^2= ac \\ b= \sqrt{ac} \]

Answer 3

so replace a,b,c with
a, k a, k^2 a

and put those values into each of the choices and see what happens

Answer 4

logb^2=log(ac) ....

Answer 5

for choice A, replace a with a, b with k a, and c with k^2 a
you get
loga^2, logb^2, logc^2 -->
log( a^2), log( (ka)^2), log( (k^2a)^2 )

remember the rules
\[ \log(a^b) = b \log(a) \\ \log(xy) = \log(x)+ \log(y) \]

can you simplify
log( a^2), log( (ka)^2), log( (k^2a)^2 )
?

Answer 6

2loga
2logk+2loga
4a logk

it seems quite strange...

Answer 7

the third term becomes
\[ \log\left( (k^2 a)^2 \right)= 2 \log(k^2 a)= 2(\log(k^2) + \log(a))=\\
4 \log(k)+ 2\log(a)
\]

Answer 8

ohh...

Answer 9

so you have the sequence
2loga
2logk+2loga
4logk +2loga

find the difference between terms. what do you get ?

Answer 10

2logk

Answer 11

so you have a common difference of 2 logk so you have an Arith. seq.

Answer 12

for B and C, look at the problems you did yesterday, because I think we already did these.

Answer 13

B is correct

Answer 14

or, you can just work it through. replace a,b,c with a, a k and a k^2 ...

Answer 15

B is a G.P.

Answer 16

a^3=a^3
b^3=k^3 a^3
c^3=k^6 a^3

common ratio is k^3, isn't it...

Answer 17

C is also correct

Answer 18

yes, B is true, with a common ratio of k^3

b^3/a^3 is (k^3 a^3)/a^3= k^3

Answer 19

C is not true. How did you do it ?

Answer 20

4^a=4^a
4^b=4^(ka)
4^c=4^(k^2 a)

common ratio is .... i don't know how to express...

Answer 21

why not?

Answer 22

the ratios are

4^b / 4^a is 4^(b-a) using the rule
\[ \frac{a^x}{a^y} = a^x \cdot a^{-y} = a^{x-y} \]
and
4^c/4^b = 4^(c-b)

if we replace a,b,c we will see that these ratios will be different

Answer 23

oic

Answer 24

thx

Answer 25

we did a version of choice C yesterday