If the coefficients of the 5 th, 6 th and 7 th terms of the expansion of (1+x)^n are in A.P, the

step 1 In the expansion of 1 plus x the whole power n, it's given that 5th, 6th, 7th terms, that is T5,

If the coefficients of the 5 th, 6 th and 7 th terms of the...

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Key Concepts

Binomial Theorem

The binomial theorem provides a formula to expand expressions of the form (1 + x)^n into a sum of terms with binomial coefficients, allowing each term in the expansion to be expressed as a combination of n and the term index.

Binomial Coefficients

Binomial coefficients, typically denoted as C(n, k) or n choose k, are the numerical factors in the expansion of a binomial raised to a power. They represent the number of ways to choose k elements from a set of n, and are computed using factorial formulas.

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence where each term after the first is obtained by adding a fixed constant, known as the common difference, to the preceding term. When applied to coefficients, this property generates specific relationships that can be used to set up equations.

Term Indexing in Binomial Expansion

Term indexing in the binomial expansion involves identifying which coefficient corresponds to a particular term in the sequence. Typically, the (k+1)th term has the coefficient C(n, k), and understanding this indexing is critical for comparing coefficients from different terms.

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