8.5PureStretch

Binomial Estimation

When $x$ is small, the early terms of a binomial expansion dominate and the rest are tiny. Keeping just the first few terms gives a quick, accurate approximation — a clever way to estimate awkward powers by hand.

25 min Video by Zeeshan Zamurred The Binomial Expansion

Edexcel AS Level Maths: 8.5 Binomial EstimationWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the first few terms of an expansion to approximate
Choose a value of x that gives the number you want
Estimate powers such as 1.01ⁿ
Judge how good an approximation is

Why early terms dominate

In $(1 + x)^{n}$ , later terms contain higher powers of $x$ . When $x$ is small (say $0.01$ ), $x^{2}$ is tiny and $x^{3}$ is negligible — so the first two or three terms already give a very close estimate.

(1 + x)^{n} \approx 1 + n x + \frac{n ( n - 1 )}{2} x^{2} (small x)

Keep the first few terms; drop the negligible tail.

Choosing the right x

To estimate a specific power, pick the substitution that turns the bracket into the number you want. For $1.0 1^{n}$ , write it as $(1 + 0.01)^{n}$ and substitute $x = 0.01$ .

1Write

1.0 1^{10} = (1 + 0.01)^{10}

, so

n = 10

x = 0.01

2First three terms:

1 + 10 (0.01) + 45 (0.01)^{2}

= 1 + 0.1 + 0.0045

Answer

\approx 1.1045

Tip — Match the bracket to the target: 0.98ⁿ comes from (1 − 0.02)ⁿ with x = −0.02.

How accurate is it?

The approximation is only valid for $small$ $x$ , and improves as you include more terms. The dropped terms (the “error”) involve higher powers of $x$ , so the smaller $x$ is, the better the estimate.

Formula recap

(1 + x)^{n} \approx 1 + n x + \frac{n ( n - 1 )}{2} x^{2}

Approximation for small x.

1.0 1^{n} = (1 + 0.01)^{n}

Choose x to match the target.

smaller x \Rightarrow better estimate

Validity / accuracy.

Common mistakes to avoid

Using the approximation when x is large.

It is only valid for small x — large x makes the dropped terms significant.

Choosing the wrong substitution, e.g. x = 0.1 for 1.01ⁿ.

1.01ⁿ = (1 + 0.01)ⁿ, so x = 0.01.

Key takeaways

For small x, the first few terms of (1 + x)ⁿ approximate it well.
Pick x so the bracket equals your target number.
More terms and smaller x both improve accuracy.
The approximation breaks down when x is not small.

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