Question

question 3
the first five triangular numbers are shown, where n represents the number of dots in the base of the figure, and d(n) represents the total number of dots in the figure.

images of triangular dot figures with labels 1, 3, 6, 10, 15

when n = 1, there is 1 dot. when n = 2, there are 3 dots. when n = 3, there are 6 dots. notice that the total number of dots d(n) increases by n each time.
use induction to prove that ( d(n) = \frac{n(n + 1)}{2} ).

part a
prove the statement is true for n = 1.
type your answer in the box.

Explanation:

Step1: Substitute n=1 into formula

Calculate $d(1)$ using $\frac{n(n+1)}{2}$:
$\frac{1(1+1)}{2} = \frac{1\times2}{2}$

Step2: Simplify the expression

$\frac{2}{2} = 1$

Step3: Match given value

When $n=1$, the given total dots is 1, which equals the formula result.

Answer:

When $n=1$, substitute into $d(n)=\frac{n(n+1)}{2}$:
$\frac{1(1+1)}{2}=\frac{2}{2}=1$, which matches the given total number of dots for $n=1$. Thus the statement is true for $n=1$.