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Sequences & Series

Question 5 of 9

Binomial Theorem & Pascal's Triangle

Question

Use the Binomial Theorem to find the first four terms in the expansion of \((2 - x)^{7}\)

Show solution Hide solution Fully worked — 4 steps
  1. Set up the expansion

    There will be 8 terms in the expansion but we only need the first four. Use the expansion of \((a + bx)^n\) where \(a = 2\), \(b = -1\) and \(n = 7\).

  2. Write out the first four terms
    \[ (2 - x)^{7} = \binom{7}{0}(2)^{7} + \binom{7}{1}(2)^{6}(-x) + \binom{7}{2}(2)^{5}(-x)^{2} + \binom{7}{3}(2)^{4}(-x)^{3} + \cdots \]
  3. Evaluate the combinations and the indices
    \[ = (1)(128) + (7)(64)(-x) + (21)(32)(x^{2}) + (35)(16)(-x^{3}) + \cdots \]
  4. Evaluate the product
    \[ = 128 - 448x + 672x^{2} - 560x^{3} + \cdots \]

    Therefore, the first four terms of \((2 - x)^7\) are \(128 - 448x + 672x^{2} - 560x^{3}\): 128, minus 448 \(x\), plus 672 \(x\) squared, minus 560 \(x\) cubed.