Binomial Theorem for a Positive Integral Index
(x + y)n expands as the sum over r of nCr x^(n-r) yr, the term with index r being the (r+1)th term.
What is Binomial Theorem for a Positive Integral Index?
(x + y)n expands as the sum over r of nCr x^(n-r) yr, the term with index r being the (r+1)th term.
Key formula / rule: The general term is T(r+1) = nCr x^(n-r) yr.
Key points
- Derive the general term of a binomial expansion.
- Compute a specified term or coefficient in an expansion.
Common exam trap
Confusing the term number with the index r.
Definitions
- Term
Binomial Theorem for a Positive Integral Index
- Meaning
(x + y)n expands as the sum over r of nCr x^(n-r) yr, the term with index r being the (r+1)th term.
- Term
Binomial Theorem for a Positive Integral Index — explanation
- Meaning
Each term records how many × y is chosen from the n brackets, so the binomial coefficients are exactly the combination counts. Term extraction questions are then a matter of solving for the right r.
Learning objectives
Derive the general term of a binomial expansion.
Compute a specified term or coefficient in an expansion.
Formulae
- Key point
The general term is T(r+1) = nCr x^(n-r) yr.
- Key point
There are n + 1 terms and the coefficients are symmetric.
- Key point
Putting x = y = 1 sums all coefficients to 2n.
Prerequisites
MAT-U1-PC-T1-S1-C2
Common mistakes
Confusing the term number with the index r.
Ignoring sign alternation when the expansion is of (x - y)n.
Keywords
Binomial
Theorem
Positive
Integral
Practice preview
What is the coefficient of x^3 in the expansion of (1 + x)^5?…
easy
How many terms are there in the expansion of (2x + 3y)^10?…
easy
What is the general term, T_r+1, in the binomial expansion of (x + y)^n?…
easy