The Fibonacci Sequence – There is a very interesting sequence of numbers, presented by the Italian mathematician Leonardo of Pisa (aka Fibonacci), called the Fibonacci sequence. The sequence goes like this – 1. Take 1 as the 1st term of the sequence 2. Take 1 as the 2nd term of the sequence too 3. For the 3rd term, add the 1st and the 2nd term. Thus, you get 1+1 = 2 as the 3rd term of this sequence 4. For the 4th term, add the 2nd and the 3rd term. This time, you get 1+2 =3 as the 4th term …

## DO WE REALLY UNDERSTAND MULTIPLICATION?

Before you tell me “Of course!” and think that I have lost my mind, let me ask you a question. What is the tens digit of 11¹² ? Stop! Don’t take out that calculator just yet. Let me first tell you how some other adults reacted to this earlier. I was helping a group of GMAT aspirants a few weeks back and asked them the same question. They immediately tried to correct me and told me what I really wanted to know was the units digit. I told them — “You wish! But the answer is no.” After that we got into this …

## ABSOLUTE VALUES AND THE NUMBER LINE

On a number line, |x−a| denotes the distance between x and a irrespective of the signs of x and a. Case 1: x and a are positive Let’s say x = 1 and a = 3 |x-a| = |1 – 3| = |-2| = 2 The distance remains the same when the values of x and a are swapped. Case 2: x and a are negative Let’s say x = -1 and a = -3 |x-a| = |-1 + 3| = |2| = 2 The distance remains the same when the values of x and a are swapped. Case 3: …

## PERMUTATION V/S COMBINATION: WHICH ONE TO USE?

The basic difference between a combination and a permutation is that while the former is just a way of selecting something, the latter is a way of selecting as well as arranging it. Q1. In how many ways can you select 2 letters out of A, B, C and D? Here, we are only interested in choosing 2 letters. Each set of 2 different letters can be considered a combination. AB, BC, CD, AD, AC and BD are the 6 different combinations possible. But if the question were posed slightly differently, as – Q2. In how many ways can 2 letters out of …

## SOLVING INEQUALITIES USING NUMBER LINE

What does a>b mean: It means “a” is always on the right side of “b” on the number line. Also, “b” is always on the left side of “a” on the number line. Now, lets consider this inequality: 4a > 5b ⇨ a > 5b/4 ⇨ a > b + b/4 We can say that a will always be on right side of (b + b/4) Hence, when “b” is positive: -> “a” will also be positive and a > b But, when “b” is negative: -> a will always be on the right side of (b + b/4) and …

## CONSECUTIVE NUMBERS AND DIVISIBILITY (BY 3 AND 6)

On a number line every nth number is divisible by n (n, 2n, 3n….). E.g.: Every 3rd number is divisible by 3 (3, 6, 9…) Every 4th number is divisible by 4 (4, 8, 12..) As every third number is divisible by 3, any number n on the number line: -> Is either, divisible by 3 (it is of the form 3k) -> Or, leaves a remainder 1 when divided by 3 (it is of the form 3k + 1) – in this case n+2 will be divisible by 3 -> Or, leaves a remainder 2 when divided by 3 …

## CONSECUTIVE NUMBERS AND DIVISIBILITY (BY 2, 4 AND 8)

Few facts to note: 1. On the number line, odd and even numbers alternate. For example: 1 is odd, 2 is even. So, for any two consecutive number one will be odd and the other even. 2. Between two consecutive even numbers, one will be divisible by 4 and the other will only be divisible by 2 and not by 4. For example: 2 and 4 -> 2 is not divisible by 4 but 4 is. 4 and 6 -> 4 is divisible by 4 but 6 is not. Let’s consider a product of any three consecutive numbers n(n+1)(n+2) Case 1: …