1. In an arithmetic series
- the first term is 16
- the 21 st term is 24
(a) Find the common difference of the series.
(b) Hence find the sum of the first 500 terms of the series.

Step 1 ：\(a_1 = 16\)

Step 2 ：\(a_{21} = 24\)

Step 3 ：\(a_n = a_1 + (n-1)d\)

Step 4 ：\(24 = 16 + (21-1)d\)

Step 5 ：\(8 = 20d\)

Step 6 ：\(d = \frac{8}{20} = \frac{2}{5}\)

Step 7 ：\(S_n = \frac{n}{2}(a_1 + a_n)\)

Step 8 ：\(S_{500} = \frac{500}{2}(16 + a_{500})\)

Step 9 ：\(a_{500} = 16 + (500-1)\frac{2}{5}\)

Step 10 ：\(a_{500} = 16 + 499\frac{2}{5}\)

Step 11 ：\(a_{500} = 16 + 998\frac{2}{5}\)

Step 12 ：\(a_{500} = 16 + 399.2\)

Step 13 ：\(a_{500} = 415.2\)

Step 14 ：\(S_{500} = \frac{500}{2}(16 + 415.2)\)

Step 15 ：\(S_{500} = 250(431.2)\)

Step 16 ：\(S_{500} = 107800\)

Step 17 ：\(\boxed{107800}\)