Question
using pascals teiangle what is the third term in the expansion of (a+b)^5
Answers
Answered by
GPT-4o mini
To find the third term in the expansion of \((a+b)^5\) using Pascal's triangle, you first need to identify the coefficients from the triangle for the corresponding row.
The coefficients for \((a+b)^n\) can be found in the \(n\)th row of Pascal's triangle, where \(n\) is the exponent. For \(n = 5\), the coefficients are:
\[
1, 5, 10, 10, 5, 1
\]
The general term in the expansion of \((a+b)^n\) is given by:
\[
\text{Term } k = \binom{n}{k} a^{n-k} b^k
\]
Where \(\binom{n}{k}\) is the binomial coefficient, and \(k\) starts at 0.
To find the third term, we set \(k = 2\) (since we start counting terms from \(k = 0\)):
\[
\text{Term } 2 = \binom{5}{2} a^{5-2} b^{2}
\]
Calculating this:
1. The binomial coefficient \(\binom{5}{2} = 10\).
2. The term then becomes:
\[
10 a^{3} b^{2}
\]
Thus, the third term in the expansion of \((a + b)^5\) is:
\[
10 a^{3} b^{2}
\]
The coefficients for \((a+b)^n\) can be found in the \(n\)th row of Pascal's triangle, where \(n\) is the exponent. For \(n = 5\), the coefficients are:
\[
1, 5, 10, 10, 5, 1
\]
The general term in the expansion of \((a+b)^n\) is given by:
\[
\text{Term } k = \binom{n}{k} a^{n-k} b^k
\]
Where \(\binom{n}{k}\) is the binomial coefficient, and \(k\) starts at 0.
To find the third term, we set \(k = 2\) (since we start counting terms from \(k = 0\)):
\[
\text{Term } 2 = \binom{5}{2} a^{5-2} b^{2}
\]
Calculating this:
1. The binomial coefficient \(\binom{5}{2} = 10\).
2. The term then becomes:
\[
10 a^{3} b^{2}
\]
Thus, the third term in the expansion of \((a + b)^5\) is:
\[
10 a^{3} b^{2}
\]