Question

69. What is 0.9+ 0.09+ 0.009 + ... equal to? 20
(a) 1
(b) 1.01
(c) 1.001
(d) 1.1​

Answer 1

Step-by-step explanation:

This question can be contradicting if we approximate this solution because we need to in these problem or you would get a never ending trailing 9's or so......

but still see what we can do

a1 = 0.9

a2 = 0.09

a3 = 0.009

and so on.....

adding these we get

0.9 + 0.09 + 0.009 + 0.0009........

= 0.99999.......

we know that 0.999999....= 1

But how ?????

Let's say x = 0.9999.....

10x = 9.9999....

subtracting both we get

10x - x = 9.9999..... - 0.9999...

9x = 9

so x = 9/9 = 1

so 0.99999..... = 1 (This is called the Rational number proof)

This is not fake they will teach you this in school

Now that is one side, even this you may feel that it is an approximation.

Similarly,let's say

a1 = 0.9 = 1

a2 = 0.09 = 0.1

a3 = 0.009 = 0.01

and so on.....

adding we get

1 + 0.1 + 0.01 + 0.001 + .......

= 1.11111......

There can even be more complication, like we said that

0.999... = 1

Similarly we can say that

0.9 = 0.89

0.09 = 0.089

and so on

adding these will be a lot crazier and stressful

and if we do so we get

Sum = 0.89 + 0.089 + 0.0089 + .......

= 0.98888.........

Don't think I am faking all this, all of these have proofs and you will learn it in 9th or higher classes

See it can get even weirder,

like I say 0.9 = 0.89999......

and 0.09 = 0.089999.......

and so on

Adding these will be even weirder

Even these sum upto 0.999........ which could end with any number like 8 or 7 or 5 or 1 or even 0

It is very complex and you cant round this upto 1

See these can be contradicting

once you got 0.999... and then you got 1.1111... which is completely different then 0.89999.... then again 0.9999.....ending with an unknown number, etc etc

So the best possible answer could anything as you like, based on your argument

exact answer would be 0.9999..... but how long can you write 9's, so let's just say they are equal to 1

I am interested in 1, you can argue for another answer, so it's all on your perspective

Believing that it is 1

Hope you understood it........All the best