Question

Among the following four statements, the statement which is not true, for all \( n \in N \) is:

• A. \( (2n + 7) < (n + 3)^2 \)
• B. \( 1^2 + 2^2 + \dots + n^2 > \frac{n^3}{3} \)
• C. \( 3.5^{2n+1} + 2^{3n+1} \) is divisible by 23
• D. \( 2 + 7 + 12 + \dots + (5n - 3) = \frac{n(5n - 1)}{2} \)

Hint:

To validate generalized statements, always start by checking small values of \( n \) to see if they hold true.

Answer 1

The Correct Option is C

We need to determine which statement is not true for all \( n \in \mathbb{N} \). Step 1: Checking Statement (1) \[ (2n + 7)<(n + 3)^2 \] Expanding the right-hand side: \[ (2n + 7)<n^2 + 6n + 9 \] Rearrange: \[ 0<n^2 + 4n + 2 \] Since \( n^2 + 4n + 2 \) is always positive for all \( n \in \mathbb{N} \), this statement is always true. Step 2: Checking Statement (2) The sum of squares formula: \[ 1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6} \] We need to check: \[ \frac{n(n+1)(2n+1)}{6}>\frac{n^3}{3} \] Multiply both sides by 6: \[ n(n+1)(2n+1)>2n^3 \] Dividing by \( n \) (for \( n \geq 1 \)): \[ (n+1)(2n+1)>2n^2 \] Expanding: \[ 2n^2 + 3n + 1>2n^2 \] \[ 3n + 1>0 \] This is true for all \( n \geq 1 \), so the statement holds. Step 3: Checking Statement (3) We need to check whether: \[ 3.5^{2n+1} + 2^{3n+1} \] is divisible by 23 for all \( n \in \mathbb{N} \). Checking modulo 23, we analyze \( 3.5^{2n+1} \mod 23 \) and \( 2^{3n+1} \mod 23 \). Computing for small values shows counterexamples where divisibility does not hold for all \( n \), meaning this statement is not true for all \( n \). Step 4: Checking Statement (4) The given arithmetic series: \[ 2 + 7 + 12 + \dots + (5n - 3) \] is an arithmetic sum with first term \( a = 2 \), common difference \( d = 5 \), and last term \( (5n - 3) \). The sum formula: \[ S_n = \frac{n}{2} [2a + (n-1)d] \] \[ = \frac{n}{2} [2(2) + (n-1)5] \] \[ = \frac{n}{2} [4 + 5n - 5] = \frac{n}{2} (5n - 1) \] which matches the given expression, proving it is true. Conclusion: The statement that is not always true is: \[ \boxed{(3) \text{ } 3.5^{2n+1} + 2^{3n+1} \text{ is divisible by 23}} \] \bigskip