Chapter 10.5, Problem 68E

(a)

To determine

To write: The first four triangular numbers as combinations

Answer to Problem 68E

  

Explanation of Solution

Given:

The first four triangular numbers are $1,3,6,10$ .

Calculation:

The Pascal’s triangle as numbers is given. The first four triangular numbers are $1,3,6,10$ .Use the Pascal’s triangle to write the first four triangular numbers as combinations.

  

Conclusion:

Therefore,

  

(b)

To determine

To write: The explicit rule for the $nth$ triangular number $T_n$ .

Answer to Problem 68E

TheRule (1) states the first, second and the $n^{th}$ triangular number.

Explanation of Solution

Calculation:

Use the Pascal’s triangle to write an explicit rule for the $nth$ triangular number $T_n$ .the rule is follow as:

  $T_n=C{}_0,C{}_1\mathrm{......}C{}_{n+1}\mathrm{.........}(1)$

Rule (1) states the first, second and the $n^{th}$ triangular number.

Conclusion:

Therefore, the Rule (1) states the first, second and the $n^{th}$ triangular number.