Answered

Explain how you can use the terms from the binomial expansion to approximate $0.98^5$.



$\begin{array}{l}

1^{\text{st}} \text{ term} = 1 \\

2^{\text{nd}} \text{ term} = -0.1 \\

3^{\text{rd}} \text{ term} = 0.004 \\

4^{\text{th}} \text{ term} \approx 0

\end{array}$



There are $5+1=6$ terms in the binomial expansion of $(1-0.02)^5$.



The $5^{\text{th}}$ term is □.



The $6^{\text{th}}$ term is □.



The approximate value of $0.98^5$ is □.

Verified Answer :

  • Express \(0.98^5\) as \((1 - 0.02)^5\) and expand using the binomial theorem.
  • Calculate the 5th term (\(T_5\)) as \(\binom{5}{4} (1) (-0.02)^4 = 0.0000008\).
  • Calculate the 6th term (\(T_6\)) as \(\binom{5}{5} (1) (-0.02)^5 = -0.000000032\).
  • Determine the approximate value by summing the dominant terms \(T_1 + T_2 + T_3 = 1 - 0.1 + 0.004 = 0.904\).
  • Final Answer: The 5th term is \(\boxed{0.0000008}\), the 6th term is \(\boxed{-0.000000032}\), and the approximation is \(\boxed{0.904}\).

Explanation

  1. Applying the Binomial Theorem

To approximate \(0.98^5\), we express it as \((1 - 0.02)^5\). The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case, \(a=1\), \(b=-0.02\), and \(n=5\). The expansion has \(n+1=6\) terms, given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). We are given the first four terms as:

  • \(T_1 = \binom{5}{0}(1)^5(-0.02)^0 = 1\)
  • \(T_2 = \binom{5}{1}(1)^4(-0.02)^1 = -0.1\)
  • \(T_3 = \binom{5}{2}(1)^3(-0.02)^2 = 10 \cdot 0.0004 = 0.004\)
  • \(T_4 = \binom{5}{3}(1)^2(-0.02)^3 = 10 \cdot (-0.000008) = -0.00008 \approx 0\).
  1. Calculating the 5th and 6th terms

To find the 5th and 6th terms, we continue applying the binomial formula:

  • For the 5th term (\(k=4\)): \(T_5 = \binom{5}{4} (1)^1 (-0.02)^4 = 5 \cdot 0.00000016 = 0.0000008\).
  • For the 6th term (\(k=5\)): \(T_6 = \binom{5}{5} (1)^0 (-0.02)^5 = 1 \cdot (-0.000000032) = -0.000000032\).
  1. Calculating the Approximation

The approximate value of \(0.98^5\) is calculated by summing the first few significant terms of the binomial expansion. Since \(T_4, T_5,\) and \(T_6\) are extremely small compared to the first three terms, we can approximate the result using \(T_1 + T_2 + T_3\):
\(\text{Approximate value} = 1 - 0.1 + 0.004 = 0.904\). This provides a very high degree of precision for the calculation of \(0.98^5\).

Examples

In finance and engineering, binomial approximations are frequently used to estimate small percentage changes over time. For instance, if an investment value decreases by 2% each year, calculating the value after 5 years involves calculating \((0.98)^5\). Rather than performing the full multiplication, binomial expansion allows for a quick, accurate estimation by focusing on the first few terms, which represent the most significant parts of the change. This simplifies complex calculations while maintaining sufficient precision for decision-making.

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