Question

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work. Sn: 2 + 5 + 8 + . . . + ( 3n - 1) = n(1 + 3n)/2

Answer 1

S1: 2

S2:7

S3: 15

Explanation:

Sn: 2 + 5 + 8 + . . . + ( 3n - 1) = n(1 + 3n)/2

S1: Put n = 1

`= (n (1 + 3n))/(2) \\= (1 (1 + 3(1)))/(2)\\= (1 (1 + 3))/(2)\\= (1(4))/(2)\\= (4)/(2)\\= 2`

S2: Put n = 2

`= (n (1 + 3n))/(2)\\= (2 (1 + 3(2)))/(2)\\= (2 (1 + 6))/(2)\\= (2 (7))/(2)\\= (14)/(2)\\= 7`

S3: Put n = 3

\frac{n (1 + 3n)}{2}\\= \frac{3 (1 + 3(3))}{2}\\= \frac{3 (1 + 9)}{2}\\= \frac{3 (10)}{2}\\= \frac{30}{2}\\= 15[/tex]

How these are true:

S1: 2

S2: 2+ 5 = 7 (Sum of first two terms

S3: 2+5+8 = 15 (Sum of first three terms)