IP AMaths Notes (Upper Sec, Year 3-4): 10) Binomial Theorem

The binomial theorem expands \( (a + b)^n \) quickly. Identify the general term and know when to truncate.

Formulae to recall

• General term: \( T_{r+1} = \binom{n}{r} a^{n-r} b^{r} \) for integer \( n \geq 0 \).
• Greatest term near maximum when \( \frac{\left|T_{r+2}\right|}{\left|T_{r+1}\right|} \leq 1 \).
• For negative or fractional indices, use the infinite series form: \( (1 + x)^k = 1 + kx + \frac{k(k - 1)}{2!} x^2 + \cdots \) for \( |x| < 1 \).

Worked example 1 — Specific term

Find the coefficient of \( x^5 \) in \( (2 - x)^7 \).

1. General term: \( T_{r+1} = \binom{7}{r} 2^{7-r} (-x)^{r} \).
2. Power of \( x \) is \( r \). Set \( r = 5 \).
3. Substitute: \( T_6 = \binom{7}{5} 2^{2} (-x)^{5} = 21 \times 4 \times (-x^{5}) = -84 x^5 \).

Coefficient is \( -84 \).

Worked example 2 — Binomial approximation

Approximate \( (1.02)^5 \) to 5 significant figures without a calculator.

1. Write \( (1 + 0.02)^5 \).
2. Use first three terms: \( 1 + 5 imes 0.02 + \frac{5 \times 4}{2} (0.02)^{2} = 1 + 0.1 + 0.004 = 1.104 \).
3. Third-order term: \( \frac{5 \times 4 \times 3}{6} (0.02)^{3} = 10 \times 8 \times 10^{-6} = 8.0 \times 10^{-5} \).
4. Sum: \( 1.10408 \) (5 s.f.).

Try this

Determine the interval of convergence for \( (1 - 3x)^{-1/2} \) and use the first three terms to approximate \( (0.97)^{-1/2} \).