Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Understand and use the binomial expansion of (a + bx)ⁿ for positive…
Full specification wording
Sequences and series: 4.1 Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n; the notations n! and ⁿCᵣ 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 binomial expansion writes (a + bx)ⁿ as a sum of terms.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 1 of 6
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
The binomial expansion writes (a + bx)ⁿ as a sum of terms. For positive integer n the series terminates after n + 1 terms, and coefficients come from n! and ⁿCᵣ, where n! = n × (n − 1) × ... × 1 and ⁿCᵣ = n! ÷ [r!(n − r)!]. These same coefficients count combinations in binomial probability. For rational n, including negative and fractional values, the expansion continues indefinitely and is valid only when |bx/a| < 1. For example, (1 + 2x)⁻¹ expands to 1 − 2x + 4x² − ... valid for |x| < 1/2. Approximation uses the first few terms when x is small, and the validity condition ensures the infinite series converges.
What earns the marks
- Expands (a + bx)ⁿ for positive integer n using ⁿCᵣ coefficients and correct ascending powers of x.
- Evaluates n! and ⁿCᵣ correctly, including cases such as ⁵C₂ = 10 and 4! = 24.
- Links binomial coefficients to binomial probability by identifying ⁿCᵣ as the number of ways of choosing r successes from n trials.
- Extends the expansion to rational n, forming the general term with the product of descending factors divided by r!.
- States and applies the validity condition |bx/a| < 1, and uses a sufficient number of terms to approximate a value to a required accuracy.
Common mistakes and how to put them right
Treating a rational-n expansion as finite and stopping after a few terms.
Put it right: Continue the pattern and state that the series is infinite.
Omitting the validity condition or writing it as |x| < 1 regardless of a and b.
Put it right: Derive |bx/a| < 1 from the ratio of successive terms.
Miscomputing ⁿCᵣ by cancelling factorials incorrectly.
Put it right: Use ⁿCᵣ = n! ÷ [r!(n − r)!] and check with a small case.
Exam tips
- Write the general term first, then substitute r values, so the pattern and signs stay consistent.
- State the validity condition in the same line as the expansion to avoid losing that mark.
- For approximations, keep one more term than the required accuracy suggests, then round at the end.
Try the lesson
The start of the Sequences and series: 4.1 Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n; the notations n! and ⁿCᵣ 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). lesson, made from checked teaching notes for this specification point.
Have a go first
Use the binomial expansion to find the coefficient of x² in the expansion of (1 + 3x)⁵.
Listen
The binomial expansion writes (a + bx)ⁿ as a sum of terms. For positive integer n the series terminates after n + 1 terms, and coefficients come from n! and ⁿCᵣ, where n! = n × (n − 1) × ... × 1 and ⁿCᵣ = n! ÷ [r!(n − r)! These same coefficients count combinations in binomial probability. For rational n, including negative and fractional values, the expansion continues indefinitely and is valid only when |bx/a| < 1. For example, (1 + 2x)⁻¹ expands to 1 − 2x + 4x² − ... valid for |x| < 1/2. Approximation uses the first few terms when x is small, and the validity condition ensures the infinite series converges. A strong answer earns marks for these points. Expands (a + bx)ⁿ for positive integer n using ⁿCᵣ coefficients and correct ascending powers of x. Evaluates n! and ⁿCᵣ correctly, including cases such as ⁵C₂ = 10 and 4! = 24. Links binomial coefficients to binomial probability by identifying ⁿCᵣ as the number of ways of choosing r successes from n trials. Extends the expansion to rational n, forming the general term with the product of descending factors divided by r! A common slip: Treating a rational-n expansion as finite and stopping after a few terms. Instead, continue the pattern and state that the series is infinite. Exam tip: Write the general term first, then substitute r values, so the pattern and signs stay consistent.
See it
Key words
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n!
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ⁿCᵣ
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Finite versus infinite binomial series
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Validity condition
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See every definition
- n!
- The product n × (n − 1) × ... × 1, used with ⁿCᵣ to give the coefficients of the expansion.
- ⁿCᵣ
- Equal to n! ÷ [r!(n − r)!], counting the number of ways of choosing r successes from n trials in binomial probability.
- Finite versus infinite binomial series
- For positive integer n the series terminates after n + 1 terms, while for rational n the expansion continues indefinitely.
- Validity condition
- The expansion for rational n is valid only when |bx/a| < 1, which ensures the infinite series converges.
Label the diagram
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1Box 1
2Box 2
3Box 3
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Test your understanding of Sequences and series: 4.1 Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n; the notations n! and ⁿCᵣ 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).
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