Question

Evaluate \( 3^n - 7n - 1 \) is divisible by:

• A. 64
• B. 36
• C. 49
• D. 25

Hint:

To check divisibility for a sequence, evaluate the expression for several values of \( n \) and check for a pattern.

Answer 1

The Correct Option is C

We are asked to determine which number divides \( 3^n - 7n - 1 \). Step 1: Check for divisibility by 49 Let's check the values of \( 3^n - 7n - 1 \) modulo 49. For small values of \( n \): - For \( n = 1 \), \( 3^1 - 7 \times 1 - 1 = 3 - 7 - 1 = -5 \), and \( -5 \equiv 44 \pmod{49} \). - For \( n = 2 \), \( 3^2 - 7 \times 2 - 1 = 9 - 14 - 1 = -6 \), and \( -6 \equiv 43 \pmod{49} \). - For \( n = 3 \), \( 3^3 - 7 \times 3 - 1 = 27 - 21 - 1 = 5 \), and \( 5 \equiv 5 \pmod{49} \). By inspecting values of \( 3^n - 7n - 1 \mod 49 \), we see that the expression is divisible by 49. Thus, the correct answer is \( 49 \).