To find the formula for the sum of the first [tex]\( n \)[/tex] terms of positive odd numbers, let's delve into the sequence and calculate the sums step-by-step.
The sequence of the first [tex]\( n \)[/tex] positive odd numbers is:
[tex]\[ 1, 3, 5, 7, 9, \ldots, (2n-1) \][/tex]
First, let's sum up the first few terms to identify any patterns:
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
\text{Sum} = 1 = 1^2 = 1
\][/tex]
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[
\text{Sum} = 1 + 3 = 4 = 2^2 = 4
\][/tex]
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[
\text{Sum} = 1 + 3 + 5 = 9 = 3^2 = 9
\][/tex]
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[
\text{Sum} = 1 + 3 + 5 + 7 = 16 = 4^2 = 16
\][/tex]
From these steps, we observe a clear pattern: the sum of the first [tex]\( n \)[/tex] positive odd numbers equals [tex]\( n^2 \)[/tex].
To generalize this, for any integer [tex]\( n \)[/tex]:
[tex]\[
S_n = n^2
\][/tex]
Therefore, the correct formula for the sum of the first [tex]\( n \)[/tex] terms of positive odd numbers is:
[tex]\[
\boxed{S_n = n^2}
\][/tex]
Hence, the correct answer is:
[tex]\[
\text{d) } S_n = n^2
\][/tex]
