- Identify the binomial components: \(a=1\), \(b=-0.02\), \(n=5\).
- Apply the general term formula \(\binom{n}{k} a^{n-k} b^k\) for \(k=0, 1, 2, 3\).
- Calculate each binomial coefficient and power term:
- 1st term (\(k=0\)): \(\binom{5}{0} (1)^5 (-0.02)^0 = 1\)
- 2nd term (\(k=1\)): \(\binom{5}{1} (1)^4 (-0.02)^1 = -0.1\)
- 3rd term (\(k=2\)): \(\binom{5}{2} (1)^3 (-0.02)^2 = 0.004\)
- 4th term (\(k=3\)): \(\binom{5}{3} (1)^2 (-0.02)^3 = -0.00008\)
- Final values: \(\text{1st}=1, \text{2nd}=-0.1, \text{3rd}=0.004, \text{4th}=-0.00008\).
Explanation
- Identifying Variables and the Binomial Formula
The binomial theorem states that the expansion of \((a + b)^n\) is given by the formula:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is the binomial coefficient.
In the expression \((1 + (-0.02))^5\), we identify the variables as:
\(a = 1\)
\(b = -0.02\)
\(n = 5\)
The task is to evaluate the first four terms of this expansion, which correspond to \(k = 0, 1, 2, 3\).
- Calculating the First Four Terms
The first term occurs at \(k = 0\):
\[\binom{5}{0} (1)^{5-0} (-0.02)^0 = 1 \cdot 1^5 \cdot 1 = 1 \times 1 \times 1 = 1\]
The second term occurs at \(k = 1\):
\[\binom{5}{1} (1)^{5-1} (-0.02)^1 = 5 \cdot 1^4 \cdot (-0.02) = 5 \times 1 \times (-0.02) = -0.1\]
The third term occurs at \(k = 2\):
\[\binom{5}{2} (1)^{5-2} (-0.02)^2 = \frac{5 \cdot 4}{2 \cdot 1} \cdot 1^3 \cdot (0.0004) = 10 \cdot 1 \cdot 0.0004 = 0.004\]
The fourth term occurs at \(k = 3\):
\[\binom{5}{3} (1)^{5-3} (-0.02)^3 = \frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1} \cdot 1^2 \cdot (-0.000008) = 10 \cdot 1 \cdot (-0.000008) = -0.00008\]
- Conclusion
Based on the calculations above, the first four terms in the binomial expansion of \((1 + (-0.02))^5\) are:
1st term = \(1\)
2nd term = \(-0.1\)
3rd term = \(0.004\)
4th term = \(-0.00008\)
Examples
The binomial expansion is a powerful tool in finance and statistics, such as approximating compound interest or growth rates. For example, if you want to estimate the value of an investment with a \(-2\%\) interest rate over 5 periods, the binomial expansion allows you to calculate the approximate change without computing the full exponentiation directly. By looking at the first few terms, you can quickly estimate the impact of the rate on the initial principal.