Question

For all \( n \in \mathbb{N} \), the sum of \( \frac{n^5}{5} + \frac{n^3}{3} + \frac{7n}{15} \) is:

• A. a negative integer
• B. a whole number
• C. a real number
• D. a natural number

Hint:

When solving expressions involving fractions, always take the least common denominator and check whether the resulting expression is an integer for all values in the given set.

Answer 1

The Correct Option is D

We need to analyze the expression: \[ S = \frac{n^5}{5} + \frac{n^3}{3} + \frac{7n}{15} \] Step 1: Taking LCM The least common multiple (LCM) of denominators \( 5, 3, \) and \( 15 \) is \( 15 \). Rewriting the terms with a common denominator: \[ S = \frac{3n^5}{15} + \frac{5n^3}{15} + \frac{7n}{15} \] \[ S = \frac{3n^5 + 5n^3 + 7n}{15} \] Step 2: Checking divisibility for all natural numbers \( n \) Factoring out \( n \): \[ S = \frac{n(3n^4 + 5n^2 + 7)}{15} \] Since \( n \) is a natural number, we must check whether the numerator \( 3n^4 + 5n^2 + 7 \) is always divisible by \( 15 \) for all \( n \in \mathbb{N} \). For \( n = 1 \): \[ 3(1)^4 + 5(1)^2 + 7 = 3 + 5 + 7 = 15, \quad {which is divisible by 15.} \] For \( n = 2 \): \[ 3(2)^4 + 5(2)^2 + 7 = 3(16) + 5(4) + 7 = 48 + 20 + 7 = 75, \quad {which is divisible by 15.} \] For any natural \( n \), the expression remains divisible by \( 15 \), ensuring that \( S \) is always a natural number. Thus, the correct answer is a natural number.