Problem 101 - A Complete Resource Book in Mathematics for JEE Main

Question

Three consecutive binomial coefficients can never be in (A) G.P. (B) H.P. (C) A.P. (D) A.G.P.

Step-by-Step Solution

Answer

Three consecutive binomial coefficients can never be in A.P.

Step 1: Understand binomial coefficients

Binomial coefficients are numbers that appear in Pascal's Triangle and are written as \( \binom{n}{k} \), representing the number of ways to choose \( k \) items from \( n \) items without considering the order. For consecutive binomial coefficients, we consider three values: \( \binom{n}{r} \), \( \binom{n}{r+1} \), \( \binom{n}{r+2} \).

Step 2: Identify sequences involved

The problem asks whether these three binomial coefficients can form specific types of sequences: Geometric Progression (G.P.), Harmonic Progression (H.P.), Arithmetic Progression (A.P.), and Arithmetico-Geometric Progression (A.G.P.).

Step 3: Evaluate Conditions for A.P.

For an arithmetic progression (A.P.), the difference between consecutive terms is constant, i.e., \( \binom{n}{r+1} - \binom{n}{r} = \binom{n}{r+2} - \binom{n}{r+1} \). This translates to the middle term being the average of the first and third terms. In general, due to the combinatorial nature, the binomial coefficients cannot maintain this relationship.

Step 4: Check other progressions if necessary

While the focus is on A.P., reviewing other progressions helps ensure none are overlooked. G.P., H.P., and A.G.P. are more complex and generally do not apply to binomial coefficients due to their multiplicative and additive nature.

Key Concepts

Geometric Progression

A geometric progression, or G.P., is a sequence where each term is multiplied by a constant ratio to get the next term. For instance, in the sequence 2, 4, 8, each term is multiplied by 2. One way to identify a geometric progression in practice is by checking if the ratio between consecutive terms remains consistent.

• If you have three terms, say a, b, and c, in geometric progression, then the relationship is given by \( \frac{b}{a} = \frac{c}{b} \).
• This constant is called the common ratio \( r \).

In the context of binomial coefficients, considering three consecutive coefficients like \( \binom{n}{r}, \binom{n}{r+1}, \binom{n}{r+2} \), it's typically not possible for these to form a G.P. The reason lies in their structure, where unlike simple numbers, they involve factorials, introducing a level of complexity that disrupts the necessary consistent ratio for a G.P.

Arithmetic Progression

An arithmetic progression, or A.P., is a sequence where each term is derived by adding a constant difference to the preceding term. Think of a sequence like 3, 5, 7, where 2 is consistently added to each term.
Here’s how you can recognize an arithmetic progression:

• For three terms a, b, and c to be in A.P., they must satisfy \( b = \frac{a+c}{2} \).
• The difference between each term should remain constant, known as the common difference \( d \).

When applying this to binomial coefficients \( \binom{n}{r}, \binom{n}{r+1}, \binom{n}{r+2} \), unfortunately, they cannot form an arithmetic progression. This is because their nature and properties prevent maintaining a constant difference due to the way the values inflate or decline based on combinatorial calculations.

Harmonic Progression

A harmonic progression, or H.P., involves terms whose reciprocals are in arithmetic progression. This means that if you have a sequence \( a, b, c \) in H.P., then \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) will form an A.P.
For any three terms, the condition to check is:

• \( 2b = \frac{ac}{a+c} \), which results from the relationship \( \frac{1}{b} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{c} \right) \).

In binomial coefficients, \( \binom{n}{r}, \binom{n}{r+1}, \binom{n}{r+2} \), forming an H.P. adds another layer of complexity. The nature of factorial-based computations disrupts the progression needed for the reciprocals to align in this manner. Consequently, like a G.P. or A.P., an H.P. cannot be formed with consecutive binomial coefficients.