You can use m_n to denote the mth number in the nth row.
Description: See solution. Recursive Rule: 1_1=1, 1_2=1, m_n=(m-1)_(n-1)+m_(n-1)
Practice makes perfect
We want to write a recursive rule that gives the mth number in the nth row m_n in Pascal's Triangle. We will start by numbering the rows.
Let's try to write the third number in the fifth row, 3_5 or 10, using the numbers in the previous row.
\begin{aligned}
\colV{10}&= {\color{#FD9000}{4}}+{\color{#FD9000}{6}} \\
{\color{#0000FF}{3}}_{\color{#FF0000}{5}}&=2_4+3_4
\end{aligned}
We see that 10 is equal to the sum of the second and third terms in the previous row.
3_5=2_4+ 3_4
⇕
3_5=( 3-1)_(5-1)+ 3_(5-1)
As such, for any term m_n in Pascal's Triangle, we can write the following.
m_n=( m-1)_(n-1)+ m_(n-1)
Note that this recursive equation is true for the rows other than the first row. We can consider it as the initial row. Now we can write a recursive rule for the numbers in Pascal's Triangle.
Recursive Rule:
1_1=1, 1_2=1, m_n=(m-1)_(n-1)+m_(n-1)
