What is the sum of the infinite sequence {0.4, 0.04, 0.004.........}

There was a typo. Edited it. Thanks

S = [0.4+0.04+0.004+0.00004+....]

0.1S = [0.04+0.004+0.0004+.......]

Subtracting both equations, we get

0.9S = 0.4

S = 4/9

What is the sum of the infinite sequence {0.4, 0.04, 0.004, .........}?
S= 0.4+ 0.04+ 0.004 +......... upto infinity
or, S/4= 0.1+ 0.01+ 0.001 +......... upto infinity
or, S/4= 0.1/(1-0.1)
or, S/4=1/9
or, S= 4/9

correct answer C


NOTE : The sum of infinite terms of a GP series S∞= a/(1-r) where 0< r<1. If a is the first term, r is the common ratio.

There is a short way to convert the recurring decimal to fraction i.e. divide the number with as many 9s as the the recurring number.
Say the number is .77777777...
So, .777777777.... can be written in fraction form as \(\frac{7}{9}\)

This is how you arrive at this "formula"
X = .7777777...
10X =7.7777777......

10X-X = (7.7777777....)-(.77777777....)

9X = 7
X = \(\frac{7}{9}\)

Now, to the question:

Say S is the sum of this infinite series.

If we notice carefully the series adds up to form a number with a recurring decimal.
S= .44444444......
We, now, just have to find what .444444444...... is in fraction form.
As we saw above this is nothing but \(\frac{4}{9}\)

Option C

1/9=0.11111111

Whatever digit replaces the numerator that's what the digit will be after the decimal. Example 5/9 will be 0.555555555

Now We have .4+.04+.004 which is nothing but 0.4444444

So thats 4/9

Answer C

The GMAT can't ask you to sum an infinite series, because to do that, you need to first prove the sum 'converges', and that requires calculus, which is beyond the scope of the test.

Nor can the GMAT ask you to sum a sequence presented this way: 0.4, 0.04, 0.004, ... because then you're left to guess what the sequence is. There is an infinite number of potential sequences beginning with those three values. For example, the sequence could be defined by a rule whereby the nth term is given by

(n-2)(n-3)(0.2) + (n-1)(n-3)(-0.04) + (n-1)(n-2)(0.002)

Then the first three terms are 0.4, 0.04, 0.004, but the fourth term is not 0.0004; It is 0.292. You can never be left guessing what the next term in a sequence is on the GMAT, and the test never asks you to 'predict' what a sequence is only from a few terms. If you need to find the terms in a sequence to answer a GMAT question, the question always must provide a rule or definition of the sequence that lets you find those terms with no possibility of dispute.

If this question is simply asking what 0.4444.... is as a decimal, then since 1/3 = 0.33333...., and 1/9 is just one third of 1/3, then 1/9 = 0.11111..... Since 4/9 is 4 times as big as 1/9, it must be that 4/9 = 0.444444... so that is the answer.