Binomial Estimation

This lesson focuses on using the binomial expansion to approximate numerical values, determine accuracy, and establish ranges of validity. This is a common exam topic in Edexcel A-Level Mathematics (9MA0), often appearing as a multi-part question.

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Recap: The General Binomial Expansion

For rational n and |x| < 1:

(1 + x)^n = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ...

For (a + bx)^n: factor out a^n first to get a^n(1 + bx/a)^n, valid for |bx/a| < 1.

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Approximating Square Roots

Worked Example 1

Use a binomial expansion to find an approximation for sqrt(1.02), stating the expansion used.

sqrt(1.02) = (1 + 0.02)^(1/2)

Using (1 + x)^(1/2) ≈ 1 + (1/2)x - (1/8)x² + (1/16)x³

With x = 0.02:

≈ 1 + (1/2)(0.02) - (1/8)(0.0004) + (1/16)(0.000008)

= 1 + 0.01 - 0.00005 + 0.0000005

= 1.0099505

Calculator: sqrt(1.02) = 1.00995049...

The approximation is accurate to 6 decimal places.

Worked Example 2

Use a suitable binomial expansion to estimate sqrt(26).

sqrt(26) = sqrt(25 x (1 + 1/25)) = 5 x (1 + 0.04)^(1/2)

(1 + 0.04)^(1/2) ≈ 1 + (1/2)(0.04) - (1/8)(0.04)² + (1/16)(0.04)³