A geometric sequence has first term 3 and common ratio 2. Find the sum of the first 8 terms, then find the 8th term.
["Geometric Sequence: Sum of First 8 Terms and the 8th Term", "A geometric sequence is a fundamental concept in mathematics, commonly used in various fields like finance, science, and computer algorithms. Understanding how to compute key properties—such as the sum of the first n terms and specific term values—enhances problem-solving skills and strengthens foundational algebra knowledge. In this article, we’ll explore a geometric sequence with a first term of 3 and a common ratio of 2, calculating the sum of the first 8 terms and identifying the 8th term.", "---", "### What Is a Geometric Sequence?", "A geometric sequence is defined by a starting term (first term) ( a ) and a constant ratio ( r ) between consecutive terms. Each term is found by multiplying the previous term by ( r ). The general formula for the ( n )-th term is:", "[\na_n = a \cdot r^{n-1}\n]", "Where:\n- ( a ) = first term\n- ( r ) = common ratio\n- ( n ) = term position", "---", "### Given Values", "For this sequence:\n- First term ( a = 3 )\n- Common ratio ( r = 2 )\n- Number of terms ( n = 8 )", "---", "### Step 1: Calculate the Sum of the First 8 Terms", "The formula for the sum ( S_n ) of the first ( n ) terms of a geometric sequence is:", "[\nS_n = a \cdot \frac{r^n - 1}{r - 1} \quad \ ext{(for ( r <br/>\ne 1 ))}\n]", "Substitute ( a = 3 ), ( r = 2 ), and ( n = 8 ):", "[\nS_8 = 3 \cdot \frac{2^8 - 1}{2 - 1} = 3 \cdot (256 - 1) = 3 \cdot 255 = 765\n]", "So, the sum of the first 8 terms is 765.", "---", "### Step 2: Find the 8th Term", "Using the formula for the ( n )-th term:", "[\na_8 = a \cdot r^{8-1} = 3 \cdot 2^7 = 3 \cdot 128 = 384\n]", "Thus, the 8th term is 384.", "---", "### Why This Matters", "Being able to compute both the sum and individual terms of a geometric sequence helps in modeling exponential growth—such as population growth, compound interest, or geometric patterns in nature. Whether you're analyzing data or solving algorithmic problems, mastering these formulas empowers you with essential mathematical tools.", "---", "### Summary", "- First term: 3\n- Common ratio: 2\n- Sum of first 8 terms: 765\n- 8th term: 384", "Use the geometric series formulas:", "[\nS_n = a \dfrac{r^n - 1}{r - 1}, \quad a_n = a \cdot r^{n-1}\n]", "to confidently solve such problems.", "---", "Keywords: geometric sequence, sum of geometric series, 8th term formula, exponential growth, algebraic sum calculation, mathematical sequences."]
