\( a_3 = 3a_2 + 4 = 3 \times 10 + 4 = 34 \)

["# Understanding the Recurrence ( a_3 = 3a_2 + 4 = 34 ): A Step-by-Step Breakdown", "Learning sequences defined by recurrence relations is essential in mathematics, computer science, and algorithm analysis. One such interesting recurrence is ( a_3 = 3a_2 + 4 ), with a derived result indicating ( a_3 = 34 ). In this article, we explore this recurrence step by step, explain how to compute values, and uncover the underlying pattern for sequences governed by such rules.", "## What is a Recurrence Relation?", "A recurrence relation defines a sequence where each term is defined as a function of one or more previous terms. For example, ( a_{n} = c \cdot a_{n-1} + d ) is a linear nonhomogeneous recurrence relation, commonly used to model recursive processes.", "The example at hand follows:\n[\na_3 = 3a_2 + 4\n]", "We are told ( a_3 = 34 ), so we can solve for ( a_2 ):", "[\n34 = 3a_2 + 4 \implies 3a_2 = 30 \implies a_2 = 10\n]", "So the sequence starts with ( a_2 = 10 ), and each subsequent term grows according to the rule.", "## Step-by-Step Calculation for ( a_3 = 34 )", "Let’s trace how ( a_2 = 10 ) leads to ( a_3 = 34 ):", "- Start with initial term: ( a_2 = 10 )\n- Apply recurrence:\n [\n a_3 = 3a_2 + 4 = 3(10) + 4 = 30 + 4 = 34\n ]", "Thus, the prediction checks out. This kind of recurrence highlights how multiplying by a constant and adding a fixed constant shapes the sequence.", "## Pattern Analysis and General Formula", "To understand more deeply, solve for a general closed-form expression of ( a_n ). The recurrence:\n[\na_n = 3a_{n-1} + 4\n]\nis a first-order linear nonhomogeneous recurrence.", "### Homogeneous solution", "The associated homogeneous recurrence is:\n[\na_n^{(h)} = 3a_{n-1}^{(h)}\n]\nSolution: ( a_n^{(h)} = A \cdot 3^n )", "### Particular solution", "Try a constant solution ( a_n^{(p)} = C ):", "[\nC = 3C + 4 \implies -2C = 4 \implies C = -2\n]", "### General solution", "[\na_n = a_n^{(h)} + a_n^{(p)} = A \cdot 3^n - 2\n]", "Use initial condition to find ( A ). Suppose ( a_1 = A \cdot 3^1 - 2 = 3A - 2 ). But we only know ( a_2 = 10 ), so set ( n = 2 ):", "[\na_2 = A \cdot 3^2 - 2 = 9A - 2 = 10 \implies 9A = 12 \implies A = \frac{4}{3}\n]", "Thus, the general formula is:\n[\na_n = \frac{4}{3} \cdot 3^n - 2 = 4 \cdot 3^{n-1} - 2\n]", "Check ( n = 2 ):\n[\na_2 = 4 \cdot 3^{1} - 2 = 12 - 2 = 10 \quad \ ext{✓}\n]", "Check ( n = 3 ):\n[\na_3 = 4 \cdot 3^{2} - 2 = 4 \cdot 9 - 2 = 36 - 2 = 34 \quad \ ext{✓}\n]", "## Real-World Applications", "Recurrence relations like ( a_n = 3a_{n-1} + 4 ) model exponential growth with added constraints — useful in:", "- Algorithm analysis: Counting nodes in ternary trees with dependencies\n- Finance: Compound interest with fixed added deposits\n- Population dynamics: Modeling population growth with constant birth rates and fixed migration", "## Conclusion", "The simple recurrence ( a_3 = 3a_2 + 4 ) leads to ( a_3 = 34 ) when ( a_2 = 10 ), illustrating how linear recursive forms evolve. Understanding this step-by-step enables deeper insight into sequences governed by such rules and supports applications in diverse fields. Whether studying sequences or optimizing algorithms, mastering recurrences is a powerful skill.", "---", "Keywords: recurrence relation, ( a_3 = 3a_2 + 4 ), sequence generation, linear recurrence, closed-form formula, algorithm analysis, mathematical patterns.\nMeta Description: Explore the recurrence ( a_3 = 3a_2 + 4 ) and how it yields 34. Learn how to compute terms, derive closed-form expressions, and apply such relations in real-world problems."]