A sequence starts with 5, and each subsequent term is 3 times the previous term plus 2. What is the 5th term?

["The Mathematical Sequence That Starts with 5 and Grows by Tripling and Adding 2 – Find the 5th Term", "Mathematics is full of fascinating patterns, and one particularly captivating sequence begins with the number 5, where each term is defined by a simple yet powerful rule: each subsequent term equals three times the previous term plus 2. This recursive sequence offers a clear entry point for learners and math enthusiasts alike, inviting exploration of exponential growth, recursion, and pattern recognition. In this article, we’ll analyze the sequence step by step, derive a formula, and calculate the 5th term with ease.", "---", "### Understanding the Rule", "The sequence starts with:", "- ( a_1 = 5 )", "From there, each term is generated using the formula:", "[\na_{n} = 3 \cdot a_{n-1} + 2\n]", "This means every new term depends fully on the prior one, making it a classic example of a non-homogeneous linear recurrence relation.", "---", "### Calculating the First Five Terms", "Let’s compute each term systematically:", "1. First term (given):\n[\na_1 = 5\n]", "2. Second term:\n[\na_2 = 3 \cdot a_1 + 2 = 3 \cdot 5 + 2 = 15 + 2 = 17\n]", "3. Third term:\n[\na_3 = 3 \cdot a_2 + 2 = 3 \cdot 17 + 2 = 51 + 2 = 53\n]", "4. Fourth term:\n[\na_4 = 3 \cdot a_3 + 2 = 3 \cdot 53 + 2 = 159 + 2 = 161\n]", "5. Fifth term:\n[\na_5 = 3 \cdot a_4 + 2 = 3 \cdot 161 + 2 = 483 + 2 = 485\n]", "---", "### The 5th Term Is: 485", "So, the fifth term in this sequence is:", "[\n\boxed{485}\n]", "---", "### The Hidden Pattern and Formula", "While calculating terms manually works for small (n), it’s useful to find a closed-form expression. The recurrence ( a_n = 3a_{n-1} + 2 ) can be solved using standard techniques for linear recurrences. The general solution is:", "[\na_n = A \cdot 3^{n-1} + B\n]", "By solving the recurrence (with initial value ( a_1 = 5 )), we find:", "[\na_n = 2 \cdot 3^{n-1} + 3\n]", "For ( n = 5 ):", "[\na_5 = 2 \cdot 3^{4} + 3 = 2 \cdot 81 + 3 = 162 + 3 = 165 \quad \ ext{(Wait! This contradicts earlier result)}\n]", "Wait — correction: The closed-form formula derived via standard solving methods is actually:", "Actually, solving the recurrence:", "Let ( b_n = a_n + c ) eliminate the constant. Try ( b_n = a_n + k ):", "[\na_n = 3a_{n-1} + 2 \Rightarrow b_n - k = 3(b_{n-1} - k) + 2\n]\n[\nb_n - k = 3b_{n-1} - 3k + 2\n]\n[\nb_n = 3b_{n-1} - 3k + 2 + k = 3b_{n-1} - 2k + 2\n]", "To remove the constant: set ( -2k + 2 = 0 \Rightarrow k = 1 )", "Then:\n[\nb_n = 3b_{n-1}, \quad b_1 = a_1 + 1 = 6\n]\nSo:\n[\nb_n = 6 \cdot 3^{n-1}\n\Rightarrow a_n = b_n - 1 = 6 \cdot 3^{n-1} - 1\n]", "Let’s verify:", "- ( n = 1 ): ( 6 \cdot 3^0 - 1 = 6 - 1 = 5 ) ✅\n- ( n = 2 ): ( 6 \cdot 3^1 - 1 = 18 - 1 = 17 ) ✅\n- ( n = 3 ): ( 6 \cdot 9 - 1 = 54 - 1 = 53 ) ✅\n- ( n = 4 ): ( 6 \cdot 27 - 1 = 162 - 1 = 161 ) ✅\n- ( n = 5 ): ( 6 \cdot 81 - 1 = 486 - 1 = 485 ) ✅", "This confirms the correct 5th term is indeed:", "[\n\boxed{485}\n]", "---", "### Why This Sequence Matters", "This sequence exemplifies how simple rules can generate rapidly growing numbers. It appears in modeling exponential growth with additional constant injection — relevant in computer science (heuristic algorithms), finance (compound interest with add-ons), and population dynamics. Additionally, studying such sequences builds foundational skills in recursion and algorithmic thinking.", "---", "### Final Takeaway", "Starting with 5 and applying the rule “multiply by 3, then add 2,” we’ve traced the sequence step by step to find:", "- ( a_1 = 5 )\n- ( a_2 = 17 )\n- ( a_3 = 53 )\n- ( a_4 = 161 )\n- ( a_5 = 485 )", "The 5th term is 485, and its derivation reveals elegant patterns underlying recursive growth.", "For anyone interested in math, cryptography, or algorithm design, mastering sequences like this is a rewarding step toward deeper analytical thinking.", "---", "Related searches:\n- How to solve recursive sequences\n- Exponential growth with offsets\n- Recursion and closed-form formulas\n- Math sequences from 1 to 100 with rules", "Start small, ask “what if?”, and explore the logic — the next number might surprise you."]