GMAT Math : Recursion

The GMAT Quantitative section may throw a curveball in the form of a recursive expression as you head towards a Q50/51.

Let’s start with an example

What is the value of √(2+√(2+√(2+√(2+….?

A. 2

B. 1

C. 3

D. 4

E. 5

Trick : The way to solve recursive relation is to form a ‘self relation’.

Let’s see how this works in practise :

Let us assume y = √(2+√(2+√(2+√(2+….

Then, y = √(2+y)

As the portion after the first ‘2’ : √(2+√(2+√(2+….  is also y

Take a minute to think through this 🙂

So, y = √(2+y

y2 = 2 + y

y2 – y – 2 = 0

For a quadratic equations, ay2 + by + c = 0, the solutions are

y  = (-b + √(b2 – 4ac))/2a and y = (-b – √(b2 – 4ac))/2a

Using the above, the solutions to y2 – y – 2 = 0 are :

y = (1 + √(1+8))/2 and (1 – √(1+8))/2 = 2 and -1

We go with option (A)

Here’s another question:

What the value of 1 + 1/(1+/(1+/(1+/..)

Readers are encouraged to take some time to practice their newly acquired skill 🙂

If y = 1 + 1/(1+/(1+/(1+/..)

y= 1 + 1/y

y2 = 1 + y

y2 – y – 1 = 0

y = (1 + √5)/2 and (1-√5)/2

The positive number (1 + √5)/2 is known as the Golden ration and is approximately around 1.618:1

The golden ratio has wide range of applications all the way from architecture to paintings in setting dimensions. Leonardo da Vinci apparently used it for some parts of the Mona Lisa 🙂