A sequence is defined by \( a_n = 2a_{n-1} + 3 \) with \( a_1 = 1 \). What is \( a_5 \)?

["A Deep Dive into Linear Recursion: Solving ( a_n = 2a_{n-1} + 3 ) with ( a_1 = 1 ) and Finding ( a_5 )", "Understanding recursive sequences is fundamental in mathematics, computer science, and algorithm design. One common pattern is linear recurrence relations, where each term depends linearly on previous terms. In this article, we’ll explore the recurrence:", "[\na_n = 2a_{n-1} + 3, \quad \ ext{with } a_1 = 1\n]", "We’ll break down the concept, derive a closed-form formula, and compute ( a_5 ) step-by-step — proving not only how to calculate this term but also enhancing your grasp of recursive sequences.", "---", "### What Is a Recursive Sequence?", "A recursive sequence defines each term using a rule based on prior terms. For example, Fibonacci numbers rely on sum of prior two terms, but here the rule is multiplicative and additive: multiplying by 2 and adding 3.", "This particular recurrence is non-homogeneous because of the constant term (+3). Solving such recursions generally involves two steps: finding the homogeneous solution and a particular solution.", "---", "### Step 1: Solve the Homogeneous Part", "The homogeneous version ignores the constant:", "[\na_n^{(h)} = 2a_{n-1}^{(h)}\n]", "This is a simple geometric sequence. Its solution is:", "[\na_n^{(h)} = C \cdot 2^n\n]", "where ( C ) is a constant determined by initial conditions.", "---", "### Step 2: Find a Particular Solution", "Because the full recurrence includes a constant term (3), we guess a constant particular solution ( a_n^{(p)} = A ).", "Substitute into the recurrence:", "[\nA = 2A + 3 \quad \Rightarrow \quad -A = 3 \quad \Rightarrow \quad A = -3\n]", "So, the particular solution is ( a_n^{(p)} = -3 ).", "---", "### Step 3: General Solution", "Combine homogeneous and particular solutions:", "[\na_n = a_n^{(h)} + a_n^{(p)} = C \cdot 2^n - 3\n]", "---", "### Step 4: Apply the Initial Condition", "Use ( a_1 = 1 ) to solve for ( C ):", "[\n1 = C \cdot 2^1 - 3 \quad \Rightarrow \quad 1 = 2C - 3 \quad \Rightarrow \quad 2C = 4 \quad \Rightarrow \quad C = 2\n]", "Thus, the closed-form solution is:", "[\na_n = 2 \cdot 2^n - 3 = 2^{n+1} - 3\n]", "---", "### Step 5: Compute ( a_5 )", "Plug ( n = 5 ) into the closed form:", "[\na_5 = 2^{5+1} - 3 = 2^6 - 3 = 64 - 3 = 61\n]", "---", "### Verification Using Recursion", "Let’s double-check by computing terms step-by-step:", "- ( a_1 = 1 )\n- ( a_2 = 2(1) + 3 = 5 )\n- ( a_3 = 2(5) + 3 = 13 )\n- ( a_4 = 2(13) + 3 = 29 )\n- ( a_5 = 2(29) + 3 = 61 )", "Consistent with our formula.", "---", "### Why This Matters", "Understanding such recursive patterns helps in algorithm analysis—especially in recurrences common in divide-and-conquer algorithms, dynamic programming, and financial modeling. Mastering closed-form solutions reduces computation overhead and provides deep insight into growth behavior.", "---", "### Final Answer", "[\n\boxed{a_5 = 61}\n]", "Whether you derive the formula algebraically or compute recursively, knowing how to find ( a_5 ) in the sequence defined by ( a_n = 2a_{n-1} + 3 ), ( a_1 = 1 ), empowers your mathematical and computational thinking.", "---", "Keywords: recursive sequence, linear recurrence, ( a_n = 2a_{n-1} + 3 ), closed-form solution, ( a_5 ) calculation, mathematics education, algorithm design.", "---", "Efficiently solving ( a_5 = 61 ) isn’t just a calculation — it’s a gateway to mastering recursive logic and its applications."]