The sum of the first n terms of a geometric sequence is given by \( S_n = 3^n - 1 \). Find the 5th term of the sequence.

["Understanding the Sum of a Geometric Sequence: Given ( S_n = 3^n - 1 ), Find the 5th Term", "Geometric sequences are foundational in mathematics, offering deep insights into exponential growth, finance, physics, and computer science. A key property that helps analyze these sequences is the formula for the sum of the first ( n ) terms, denoted ( S_n ). In this article, we explore a specific geometric sequence where:", "[\nS_n = 3^n - 1\n]", "We’ll derive how this formula determines the 5th term of the sequence, providing clear, practical understanding perfect for students and learners studying sequences and series.", "---", "### What is a Geometric Sequence?", "A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio ( r ). The general form is:", "[\na, ar, ar^2, ar^3, \dots\n]", "The sum of the first ( n ) terms is usually given by:", "[\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(for } r <br/>\ne 1\ ext{)}\n]", "However, in this problem, the sum is given directly as ( S_n = 3^n - 1 ), which reveals a special pattern that allows us to bypass general formulas and directly compute individual or specific terms — including the elusive 5th term.", "---", "### Why ( S_n = 3^n - 1 ) Represents a Geometric Sequence", "Given ( S_n = 3^n - 1 ), we test small values of ( n ) to verify:", "- For ( n = 1 ):\n ( S_1 = 3^1 - 1 = 2 ) ⇒ First term ( a = 2 )", "- For ( n = 2 ):\n ( S_2 = 3^2 - 1 = 9 - 1 = 8 )\n Since ( S_2 = a + ar ), and ( a = 2 ),\n ( 2 + 2r = 8 \Rightarrow 2r = 6 \Rightarrow r = 3 )", "- For ( n = 3 ):\n ( S_3 = 3^3 - 1 = 27 - 1 = 26 )\n ( S_3 = a + ar + ar^2 = 2 + 6 + 18 = 26 ), which checks out", "With ( a = 2 ) and ( r = 3 ), the sequence begins:\n( 2, 6, 18, 54, 162, \dots ) — clearly geometric with ratio 3.", "Thus, the sum formula uniquely corresponds to a geometric sequence with first term ( a = 2 ) and common ratio ( r = 3 ).", "---", "### Finding the ( n )th Term from the Sum Formula", "While the sum formula describes cumulative totals, the individual terms follow the geometric progression rule:", "[\na_n = a \cdot r^{n-1}\n]", "We already have:\n- ( a = 2 )\n- ( r = 3 )", "So,", "[\na_n = 2 \cdot 3^{n-1}\n]", "The 5th term corresponds to ( n = 5 ):", "[\na_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162\n]", "---", "### Alternative: Using the Sum Definition to Find ( a_5 )", "Another elegant method is to compute ( a_5 ) directly from the sum formula:", "[\nS_n = 3^n - 1\n]", "Note that the ( n )th term is:", "[\na_n = S_n - S_{n-1}\n]", "So,", "[\na_5 = S_5 - S_4 = (3^5 - 1) - (3^4 - 1) = 243 - 1 - (81 - 1) = 242 - 80 = 162\n]", "This confirms our earlier result using the closed-form formula.", "---", "### Why This Formula Matters (Applications)", "Understanding how the sum formula ( S_n = 3^n - 1 ) links to individual terms reveals powerful ideas:", "- It models exponential growth — such as compound interest or bacterial population with consistent ratios\n- It shows how cumulative data in sequences ties directly to term values\n- It supports problem-solving in algorithmic complexity, where geometric growth determines runtime or resource usage", "---", "### Summary", "Given that the sum of the first ( n ) terms of a geometric sequence is:", "[\nS_n = 3^n - 1\n]", "We deduce:\n- First term: ( a = 2 )\n- Common ratio: ( r = 3 )\n- The ( n )th term: ( a_n = 2 \cdot 3^{n-1} )", "For the 5th term:", "[\n\boxed{a_5 = 162}\n]", "This approach — using the sum formula to infer term values — is a valuable skill for mastering sequences in mathematics, physics, engineering, and data science.", "---", "Keywords: geometric sequence sum formula, ( S_n = 3^n - 1 ), nth term of geometric sequence, find 5th term, exponential sequences, mathematical derivation, sequences and series, exponential growth, common ratio 3, first term calculation."]