Which sum is equal to $$\sum_{k=1}^{20} (2k + 1)?$$ (A) 1 + 2 + 3 + 4 + \ldots + 20 (B) 1 + 3 + 5 + 7 + \ldots + 41 (C) 3 + 4 + 5 + 6 + \ldots + 20 (D) 3 + 5 + 7 + \ldots + 41 - HSC - SSCE Mathematics Extension 1 - Question 1 - 2016 - Paper 1

We start with the summation:

$\sum_{k=1}^{20} (2k + 1)$

This can be simplified as follows:

$\sum_{k=1}^{20} (2k + 1) = \sum_{k=1}^{20} 2k + \sum_{k=1}^{20} 1$

Calculating each part:

1. The sum of the first 20 integers:
   $\sum_{k=1}^{20} k = \frac{20(20 + 1)}{2} = 210$ Thus,
   $\sum_{k=1}^{20} 2k = 2 \times 210 = 420$

2. The sum of 20 ones:
   $\sum_{k=1}^{20} 1 = 20$

Combining both results, we have:

$\sum_{k=1}^{20} (2k + 1) = 420 + 20 = 440$