Question

Please help me :)


If x, y, z are in A.P. and tan−1x, tan−1y and tan−1z are also in A.P., then

1. 2x = 3y = 6z
2. 6x = 3y = 2z
3. 6x = 4y = 3z
4. x = y = z

Answer 1

Ahoy Matey! What are the directions for the problem?

Answer 2

Just to make sure, what is A.P. ?

Answer 3

arithematic progression

Answer 4

Ok, thx matey! My brain was goin' nuts there ok so do you remember the definition of an Arithmetic Progression?

Answer 5

well
how do i explain now :)

Answer 6

Let me write an example so it can be clearer as as sometimes definitions are hard to understand...

2,5,8 <--- these numbers are in arithmetic progression, as the difference between them (consecutives) are always equal - let's check it

5-2 = 3
8-5 = 3
And that is called the common difference

If you have 3 numbers in A.P. you can write a relation between them
as they have the common difference, in our case we do have x, y and z,
what can you say about them if they are in A.P. ?

Answer 7

well
tbh i know that
thats for explaining

Answer 8

\[t _{n}=a+(n-1)d\]

Answer 9

a=first term
d= common difference
n= term

Answer 10

Z – y
Y – x
we do not know what is the common difference equal to
Z – y = y – x <---- but what we do know is that it is equal to both expressions
And we can work a little bit with it. Let’s isolate y, then add x at both sides so we can isolate 2y And we get x + z = 2y

Answer 11

\[\tan^{-1} x ,\tan^{-1} y,\tan^{-1} z\]

Answer 12

wait i can see question marks
what did u type

Answer 13

Refresh page matey its a odd glitch

Answer 14

\(\color{blue}{\text{Originally Posted by}}\) @rvc
\[t _{n}=a+(n-1)d\]
\(\color{blue}{\text{End of Quote}}\)
That is the formula for the general term but we do not need to use it
as we just have 3 terms and we found the relation for them using the basic principles

Answer 15

yes
i know
i just made u sure that i know A.P :)

Answer 16

Hehe matey, I know, I know! And now would you please write the same relation for tan^-1x , tan^-1y and tan^-1z

Answer 17

well those are inverse functions okay:)

Answer 18

ah
im lagging
sorry

Answer 19

Same here! The lag is real. ;-; Anyhow, we do have tan^-1 x tan^-1 y and tan^-1 z, we can write the same expression for those terms... would ye please write it?

Answer 20

We just need to start with the double of the middle term, what is the middle term?

Answer 21

\[\tan^{-1} y-\tan^{-1} x ,\tan^{-1} z-\tan^{-1} y\]
did u mean this :)

Answer 22

Ye! (^: nd then solve it for 2tan^-1(y)

Answer 23

\[2\tan^{-1} y=\tan^{-1} x+\tan^{-1} z\]

Answer 24

That's absolutely r8 matey!!! Ye ho

Answer 25

may i know what's matey?

Answer 26

Ehehehe, it's a word often used by pirates to address friends (;

Answer 27

cool :)

Answer 28

Ye ho! Also remember please
ALWAYS!: if 3 terms are in A.P. then the double of the middle term is equal to the sum of the other 2 terms and you can see it in both expression 2 times the middle term
equal to the sum of the other terms

and the only solution for this system of equation is when all thre are equal to each other
in other words> x = y = z

Answer 29

..And... let's verify it, if x = y and z = y
would you please plug it?

Answer 30

hmm..
so what should i do?

Answer 31

You just need to verify it as x = y and z = y, plug instead of x y and instead of z y
to verify we got it correctly

Answer 32

oh

Answer 33

oh
the RHS =LHS

Answer 34

you just need to verify the solution as x = y
you plug instead of x just y and the same for z so you can see that it is correct
right hand = left hand, yes matey

Answer 35

so the correct ans is
x=y=z
@Miracrown thank you so much :)

Answer 36

thank you for giving your time @Miracrown
You helped me a lot :)

Answer 37

Ye, that's right on the bat! And no worries matey! (^: