Answer:
To find the third term in the expansion of \((a^2 + b)^4\) using Pascal's triangle, we first need to identify the coefficients from the fourth row of Pascal's triangle, where the row indices start from 0.
The fourth row of Pascal's triangle is: 1 4 6 4 1
Now, let's write out the expansion of \((a^2 + b)^4\):
\((a^2)^4 + 4(a^2)^3b + 6(a^2)^2b^2 + 4(a^2)b^3 + b^4\)
Simplify the terms:
\(a^8 + 4a^6b + 6a^4b^2 + 4a^2b^3 + b^4\)
The third term in this expansion is the coefficient of \(a^4b^2\), which corresponds to the third term in the fourth row of Pascal's triangle, which is 6.
So, the third term in the expansion of \((a^2 + b)^4\) is \(6a^4b^2\).
