D: Sequences and series: Understand and use the binomial expansion of a + bx n for positive integer n ; the notations n! , nCr and n r ; link to binomial probabilities. Extend to any rational n , including its use for approximation; be aware that the expansion is valid for bx a < 1 . (Proof not required.)

The idea

The binomial expansion allows you to write (a + bx)^n as a sum of terms. For positive integer n, the expansion is finite and uses binomial coefficients nCr = n!/(r!(n−r)!). These coefficients also appear in binomial probability calculations, linking algebra to statistics. For rational n (including negative and fractional), the expansion becomes an infinite series, valid only when |bx/a| < 1. You must understand the notation n!, nCr, and the binomial coefficient written as n r. To use the expansion, identify a, b, and n, then apply the formula term by term. For approximation, substitute a small value of x and truncate the series, ensuring the validity condition holds. The skill is assessed through expansion, finding specific coefficients, and using expansions to approximate values.

What earns the marks

• Correctly apply the binomial theorem for positive integer n, using nCr or factorial notation to find coefficients.
• Expand (a + bx)^n for rational n, writing the general term and simplifying coefficients, including negative and fractional powers.
• State and apply the validity condition |bx/a| < 1 for infinite expansions, and use it to determine when an approximation is valid.
• Link binomial coefficients to binomial probabilities, for example by recognising nCr as the number of ways to choose r successes in n trials.
• Use a truncated binomial expansion to approximate a numerical value, such as (1 + x)^n for small x, and estimate the error or accuracy.

Common mistakes and how to put them right

• Forgetting to include the binomial coefficient in each term.

  Put it right: Write nCr explicitly before simplifying.

• Misapplying the validity condition by using |x| < 1 instead of |bx/a| < 1.

  Put it right: Identify a and b correctly and state the condition in terms of the given expression.

• Incorrectly handling negative or fractional powers in the expansion, leading to sign errors.

  Put it right: Use the formula carefully and check signs, especially for negative n.

Exam tips

• Write the general term first; this helps you avoid mistakes and makes it easier to find a specific coefficient.
• When approximating, state the validity condition and substitute a value that satisfies it; show enough terms to achieve the required accuracy.
• Link binomial coefficients to probability by recalling that nCr gives the number of combinations, which is useful in binomial distribution questions.