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.

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 = 1(4)/2 = 2 check

S2 = 2+5 = 7 = ? 2(7)/2 = 7 check
S3 = 7+8 = 15 = ? 3 (1+9) /2 = 3*5 = 15 check

Answer 2

Thank you so much !

Answer 3

You are welcome.

Answer 4

To demonstrate that the statement Sn is true, we need to break it down into smaller statements and prove each one individually.

Let's start by expanding the left side of the equation in Sn and simplifying it:
2 + 5 + 8 + ... + (3n - 1)

The common difference between each term in the sequence is 3, so we can rewrite the terms as follows:
2 + (2 + 3) + (2 + 2(3)) + ... + (2 + (n - 1)(3))

Now we can simplify the terms inside the parentheses:
2 + 2 + 6 + 2 + 12 + ... + 2 + (n - 1)(3)

By grouping the terms, we can rewrite the expression as follows:
(2 + 2 + 2 + ... + 2) + 3(1 + 2 + 3 + ... + (n - 1))

The first part, 2 + 2 + 2 + ... + 2, is just 2 repeated n times, which can be written as 2n:
2n + 3(1 + 2 + 3 + ... + (n - 1))

Now, let's work on the second part of the expression, 1 + 2 + 3 + ... + (n - 1). This part is a sum of an arithmetic series, where the first term is 1, the common difference is 1, and the number of terms is (n - 1).

We can use the formula for the sum of an arithmetic series to find this sum:
S = (n/2)(first term + last term)

The first term is 1, and the last term is (n - 1), so we have:
1 + 2 + 3 + ... + (n - 1) = (n/2)(1 + (n - 1))

Further simplifying:
1 + 2 + 3 + ... + (n - 1) = (n/2)(n)

Now we can substitute this result back into the expression:
2n + 3(1 + 2 + 3 + ... + (n - 1)) = 2n + 3(n/2)(n)

Simplifying further:
2n + 3(n/2)(n) = 2n + (3n^2)/2

Combining like terms:
2n + (3n^2)/2 = (4n + 3n^2)/2

Now, let's simplify the right side of the equation in Sn:
n(1 + 3n)/2

Expanding and simplifying:
n + 3n^2/2

Now we can see that the left side and the right side of Sn equation are the same:
(4n + 3n^2)/2 = n + 3n^2/2

Therefore, we have shown that the statement Sn: 2 + 5 + 8 + ... + (3n - 1) = n(1 + 3n)/2 is true.