Questions: A statement Sn about the positive integers is given. Write statements SK and SK+1, simplifying statement SK+1 completely. Sn: 6+12+18+...+6n=3n(n+1) Write statement Sk. Sk: 6+12+18+...+square= square (Do not simplify.)

Solution Steps

To solve this problem, we need to express the given statement \( S_n \) for a specific integer \( k \). The statement \( S_n \) is a formula for the sum of a series. For \( S_k \), we substitute \( n \) with \( k \) in the series and the formula. The series is an arithmetic sequence with a common difference of 6, starting from 6. We will write the series up to the \( k \)-th term and equate it to the formula with \( n \) replaced by \( k \).

The statement \( S_k \) represents the sum of the first \( k \) terms of the series \( 6 + 12 + 18 + \cdots + 6k \). This can be expressed as: \[ S_k: 6 + 12 + 18 + \cdots + 6k \]

The series can be written in terms of \( k \) as: \[ S_k = 6 \cdot 1 + 6 \cdot 2 + 6 \cdot 3 + \cdots + 6 \cdot k \] This simplifies to: \[ S_k = 6(1 + 2 + 3 + \cdots + k) \]

The sum of the first \( k \) integers is given by the formula: \[ 1 + 2 + 3 + \cdots + k = \frac{k(k + 1)}{2} \] Substituting this into our expression for \( S_k \): \[ S_k = 6 \cdot \frac{k(k + 1)}{2} = 3k(k + 1) \]

Substituting \( k = 5 \) into the expression: \[ S_5 = 3 \cdot 5 \cdot (5 + 1) = 3 \cdot 5 \cdot 6 = 90 \] Thus, we have: \[ S_5: 6 + 12 + 18 + 24 + 30 = 90 \]

Final Answer