what is the value of x1

[Needgmat]

In the sequence of positive numbers x1,x2,x3,....., what is the value of x1?

1) xi = xi-1/2 for all integers i>1.

2) x5 = x4/x4 + 1

OAC

Please explain.

[crackverbal]

Hi Needgmat,

I am assuming that the 1, 2, 3 in x1, x2 and x3 are all subscripts and for statement 1, i and i-1 are again subscripts of x.

The GMAT will normally test you on two types of sequences :

1. An equally spaced sequence (having a common difference between two consecutive elements), also referred to as an Arithmetic Progression e.g. 2,4,6,8......

2. A sequence which has a common ratio or a common factor between two successive elements, also referred to as a Geometric Progression e.g. 2,4,8,16,32..., the common ratio here being 1 : 2 or the common factor being *2

Statement 1 : xi = x(i-1)/2 for all integers i > 1

Here we have a general formula connecting two consecutive terms

If we substitute i = 2 we get x2 = x1/2
If we substitute i = 3 we get x3 = x2/2 = x1/4.......

This is basically a geometric progression with common ratio 2 : 1 or common factor 1/2. However we do not have any information about the value of any terms x1, x2, x3.... Insufficient

Statement 2 : x5 = x4/(x4 + 1)

Here we have a relationship between the terms x5 and x4, but we still cannot use this information to solve for x1. Insufficient. Remember to look at this statement and not to use the information given in the first statement into the second. This is a common trap that you'll be prone to fall for if you are not careful.

Combining statements 1 and 2

Whenever you combine statements it always makes sense to substitute one statement into the other or use a mathematical operation between the two statements which leads to the target answer. This is a very important strategy that you can always apply to a DS question.

From Statement 2 we have x5 = x4/(x4+1) and from Statement 1 we can deduce that x5 = x4/2

x5 = x4/(x4+1) = x4/2 -----> x4 = 1.

Since x4 = 1 and the sequence given here is a Geometric progression having a common ratio of 2 : 1 or a common factor of 1/2, the sequence (x1, x2, x3, x4...) is (8, 4, 2, 1....). So x1 = 8 . Sufficient.

OA : C

Use the following link https://gmatonline.crackverbal.com/cours ... nt-on-gmat to learn some more strategies to solve DS questions like the one used above.

CrackVerbal Academics Team

[GMATGuruNY]

Statement 2 should appear as follows:

In the sequence of positive numbers x1, x2, x3,..., what is the value of x1?

1. xi = (xi-1) / 2 for all integers i>1
2. x5 = x4/(x4 + 1)

Statement 1: x(i) = [x(i -1)]/2.
In other words, each term is 1/2 the value of the preceding term.
Since x� can be equal to any positive value, INSUFFICIENT.

Statement 2: xâ‚… = xâ‚„/(xâ‚„ + 1)
Here, only the relationship between xâ‚… and xâ‚„ is known.
Thus, x� can be equal to any positive value.
INSUFFICIENT.

Statements combined:
Statement 1 indicates that each term is 1/2 the preceding term, implying that xâ‚… = xâ‚„/2.
Statement 2 indicates that xâ‚… = xâ‚„/(xâ‚„ + 1).

Since the two expressions in red are equal to the same value, we get:
xâ‚„/2 = xâ‚„/(xâ‚„ + 1).

Cross-multiplying, we get:
(xâ‚„)(xâ‚„ + 1) = 2xâ‚„

Since all of the values in the sequence are positive, we can safely divide each side by xâ‚„, yielding the following:
xâ‚„ + 1 = 2
xâ‚„ = 1.

Since each term is 1/2 the preceding term, we get:
x₄ = 1, x₃ = 2, x₂ = 4, x� = 8.
SUFFICIENT.

The correct answer is C.