Question

fill in the first 7 rows of pascals triangle below: speed hint: use the tab button to move between boxes! use pascals triangle to determine the following: the 3rd term of (a + b)^6 is the 2nd term of (a - b)^4 is

Explanation:

Step1: Recall binomial theorem formula

The general term of $(a + b)^n$ is $T_{r + 1}=C(n,r)a^{n - r}b^{r}$, where $C(n,r)$ is the binomial - coefficient which can be found in Pascal's triangle. The $n$ - th row of Pascal's triangle gives the coefficients of $(a + b)^{n-1}$.

Step2: Find the 3rd term of $(a + b)^6$

For $(a + b)^n$, the terms are numbered starting from $r = 0$. So for the 3rd term, $r=2$ and $n = 6$. The binomial - coefficient $C(6,2)$ from the 7th row (since we start counting rows from 1) of Pascal's triangle is 15. The 3rd term $T_{3}=C(6,2)a^{6 - 2}b^{2}=15a^{4}b^{2}$.

Step3: Find the 2nd term of $(a - b)^4$

For $(a - b)^4=(a+(-b))^4$, for the 2nd term, $r = 1$ and $n = 4$. The binomial - coefficient $C(4,1)$ from the 5th row of Pascal's triangle is 4. The 2nd term $T_{2}=C(4,1)a^{4 - 1}(-b)^{1}=-4a^{3}b$.

Answer:

The 3rd term of $(a + b)^6$ is $15a^{4}b^{2}$
The 2nd term of $(a - b)^4$ is $-4a^{3}b$