Question

Use the approach in Gauss's problem to find the sums of the arithmetic sequences below(do not use formulas).e without aa. 1+ 2+ 3+ 4 + ... +39b. 1+3+5 + 7 + ... + 2409c. 3+ 6 + 9 + ... + 300d. 1000 +995 +990 + ... + 5a. The sum of the sequence is

Answer 1

Gauss's approach consists in group highest terms with lowest terms

a. 1+ 2+ 3+ 4 + ... +39

(1 + 39) + (2+38) + (3+37) + ... + (19+21) + 20

There are 19 terms in parentheses, all equal to 40. Then, the sum is equal to:

19*40 + 20 = 780

b. 1+3+5 + 7 + ... + 2409

(1+2409) + (3+2407) + (5+2405) + ... + (1203+1207) + 1205

there are (1203+1)/2 = 602 terms equal to 2410. Then, the sum is equal to:

602*2410 + 1205 = 1452025

c. 3+ 6 + 9 + ... + 300

(3+297) + (6+294) + (9+291) + ... + (147+153) + 150 + 300

there are 147/3 = 49 terms equal to 300. Then, the sum is equal to:

49*300 + 150 + 300 = 15150

d. 1000 +995 +990 + ... + 5

1000 + (995+5) + (990+10) + ... + (505+495) + 500

the sum has 1000/5 = 200 terms, then there are 99 terms in parentheses, all of them equal to 1000. Then, the sum is equal to:

1000 + 99*1000 + 500 = 100500