Smita04 wrote:

Ax(y) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then Ax(Ax(Ax(Ax(Ax(x))))) is between

(A) 0 and ½

(B) ½ and 1

(C) 1 and 1½

(D) 1½ and 2

(E) 2 and 2½

Sorry, but I don't have the OA.

The question should read:

\(A_x(y)\) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then \(A_x(A_x(A_x(A_x(A_x(x)))))\) is between

(A) 0 and ½

(B) ½ and 1

(C) 1 and 1½

(D) 1½ and 2

(E) 2 and 2½

According to the stem:

\(A_x(x)=(x+1)x=x^2+x\);

\(A_x(A_x(x))=A_x(x^2+x)=(x^2+x+1)*x=x^3+x^2+x\);

\(A_x(A_x(A_x(x)))=A_x(x^3+x^2+x)=(x^3+x^2+x+1)*x=x^4+x^3+x^2+x\);

...

We can see the pattern now, so \(A_x(A_x(A_x(A_x(A_x(x)))))=x^6+x^5+x^4+x^3+x^2+x\).

For \(x=\frac{2}{3}\) we'll get: \((\frac{2}{3})^6+(\frac{2}{3})^5+(\frac{2}{3})^4+(\frac{2}{3})^3+(\frac{2}{3})^2+(\frac{2}{3})\).

So, we have the sum of the 6 terms of the geometric progression with the first term equal to \(\frac{2}{3}\) and the common ratio also equal to \(\frac{2}{3}\).

Now, the sum of

infinite geometric progression with

common ratio \(|r|<1\), is \(sum=\frac{b}{1-r}\), where \(b\) is the first term. So, if we had infinite geometric progression instead of just 6 terms then its sum would be \(Sum=\frac{\frac{2}{3}}{1-\frac{2}{3}}=2\). Which means that the sum of this sequence will never exceed 2, also as we have big enough number of terms (6) then the sum will be very close to 2, so we can safely choose answer choice D.

Answer: D.

One can also use direct formula.

We have geometric progression with \(b=\frac{2}{3}\), \(r=\frac{2}{3}\) and \(n=6\);

\(S_n=\frac{b(1-r^n)}{(1-r)}\) --> \(S_{6}=\frac{\frac{2}{3}(1-(\frac{2}{3})^{6})}{(1-\frac{2}{3})}=2*(1-(\frac{2}{3})^{6})\). Since \((\frac{2}{3})^{6}\) is very small number then \(1-(\frac{2}{3})^{6}\) will be less than 1 but very close to it, hence \(2*(1-(\frac{2}{3})^{6})\) will be less than 2 but very close to it.

Answer: D.

Hope it's clear.

_________________

New to the Math Forum?

Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:

GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:

PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.

What are GMAT Club Tests?

Extra-hard Quant Tests with Brilliant Analytics