a We are given Pascal's Triangle with the first four triangular numbers marked.
We are asked to write these first four triangular numbers as combinations. To do so, we will rewrite Pascal's Triangle as combinations.
Looking at the Pascal's triangle, we can write each triangular number as a combination.
Triangular Number
Combination
1
_2C_2
3
_3C_2
6
_4C_2
10
_5C_2
b We are asked to write an explicit rule for the nth triangular number T_n. Since every number can be represented as a combination, let's recall the formula for the combinations of k objects taken r at a time.
_kC_r= k!/( k- r)! r!
Looking at our result from Part A, we can see every triangular number written as a combination has r= 2. We can also see that k starts at 2, and increases by 1 every time. Let's see how we can write the first 4 triangular numbers.
n
n +1
Combination
1
1 + 1 = 2
_2C_2
2
2 + 1 = 3
_3C_2
3
3 + 1 = 4
_4C_2
4
4 + 1 = 5
_5C_2
We can see that for every value of n the value of k is n+1. Let's write the rule.
T_n = _(n+ 1)C_2
We can also write an explicit rule by substituting the formula for the combinations of n + 1 objects taken 2 at a time.
