The coefficients of three consecutive terms of (1+x)^n+5 are in the ratio 5: 10: 14. Then n=

Name: Problem 24, Chapter 8: 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics
Uploaded: 2021-11-18T14:18:03-08:00
Duration: 2 min 4 s
Channel: Aman Gupta
Description: The coefficients of three consecutive terms of (1+x)^n+5 are in the ratio 5: 10: 14. Then n=

step 1 in this problem let the coefficients of three consecutive terms be one plus x the coefficient of

The coefficients of three consecutive terms of (1+x)^n+5 are...

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (1 + x)^m, where each term in the expansion is given by a binomial coefficient multiplied by an appropriate power of x. This theorem is fundamental in combinatorics and algebra, as it connects algebraic expansions with combinatorial interpretations.

Binomial Coefficients

Binomial coefficients, typically denoted as C(m, k) or 'm choose k', represent the number of ways to select k items from a collection of m items without regard to order. They are central components of the binomial expansion and exhibit various symmetry and recursive properties that are widely used in both theoretical and applied mathematics.

Ratio of Consecutive Binomial Coefficients

The ratio of consecutive binomial coefficients can be determined by considering the relationship between terms like C(m, k) and C(m, k+1). This ratio, derived from the factorial definition of the coefficients, simplifies to an expression involving k and m, and can be set equal to a given numerical ratio to establish equations that help in finding unknown parameters.

Algebraic Manipulation in Combinatorial Equations

Solving problems involving ratios of binomial coefficients often requires algebraic manipulation of factorial expressions. By equating the simplified ratio of consecutive coefficients to a specified ratio, one can derive equations that, when solved, yield the values of unknown variables in the combinatorial setting.

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