To prove the statement [tex]\( d(n) = \frac{n(n+1)}{2} \)[/tex] is true for [tex]\( n = 1 \)[/tex]:
1. First, we need to verify the value of [tex]\( d(n) \)[/tex] when [tex]\( n = 1 \)[/tex].
2. For [tex]\( n = 1 \)[/tex], substitute [tex]\( n = 1 \)[/tex] into the formula [tex]\( d(n) = \frac{n(n+1)}{2} \)[/tex]:
[tex]\[
d(1) = \frac{1 \cdot (1+1)}{2}
\][/tex]
3. Simplify the expression:
[tex]\[
d(1) = \frac{1 \cdot 2}{2}
\][/tex]
[tex]\[
d(1) = \frac{2}{2}
\][/tex]
[tex]\[
d(1) = 1
\][/tex]
4. Therefore, for [tex]\( n = 1 \)[/tex], [tex]\( d(n) = \frac{n(n+1)}{2} \)[/tex] results in 1, which matches the given number of dots.
Thus, the statement [tex]\( d(n) = \frac{n(n+1)}{2} \)[/tex] is true for [tex]\( n = 1 \)[/tex].
