Particular Terms for Positive Integral Index

Sometimes, a particular term satisfying certain conditions is required in the binomial expansion of the type (a + b)ⁿ. This can be done by expanding (a + b)ⁿ and then locating the required term. Generally this becomes a tedious task, specially when the index n is large. In such cases, we begin by evaluating the general term T_r+1 and then finding the value of r by assuming T_r+1 to be the required term.

To get the term independent of x, we put the power of x equal to zero and get the value of r for which the term is independent of x. Putting this value of r in T_r+1, we get the term independent of x.

Working rules for finding particular terms:

Step 1:

Find the general term T_r+1 in the expansion of (a + b)ⁿ.

Step 2:

Assume that T_r+1 is the desired particular term.

Step 3:

Find the value of r.

Step 4:

Put the value of r in the term T_r+1. This gives the required particular term(s).

Example:

Find the coefficient of x⁴⁰ in the expansion of (1 + 2x + x²)²⁷ .

Suggested answer:

Let T_r+1 be the term containing x⁴⁰.

\ r = 40