A geometric sequence starts with 3 and has a common ratio of 2. What is the sum of the first 5 terms?
["Understanding a Geometric Sequence: Starting with 3 and a Common Ratio of 2", "A geometric sequence is a special type of numerical pattern where each term after the first is found by multiplying the previous term by a constant value known as the common ratio. These sequences appear in mathematics, science, finance, and computer algorithms, making them an essential concept for students and professionals alike.", "Consider the geometric sequence that begins with 3 and has a common ratio of 2. This sequence follows the rule:", "[\na_n = a_1 \ imes r^{n-1}\n]", "where:\n- (a_1 = 3) (the first term),\n- (r = 2) (the common ratio),\n- (n) is the term number.", "The first few terms are calculated as follows:\n- First term: (3)\n- Second term: (3 \ imes 2 = 6)\n- Third term: (6 \ imes 2 = 12)\n- Fourth term: (12 \ imes 2 = 24)\n- Fifth term: (24 \ imes 2 = 48)", "Now, if we want the sum of the first 5 terms, we add them together:", "[\n3 + 6 + 12 + 24 + 48 = 93\n]", "Alternatively, the sum (S_n) of the first (n) terms of a geometric sequence can be calculated using the formula:", "[\nS_n = a_1 \ imes \frac{r^n - 1}{r - 1} \quad \ ext{(for } r <br/>\ne 1\ ext{)}\n]", "Substituting (a_1 = 3), (r = 2), and (n = 5):", "[\nS_5 = 3 \ imes \frac{2^5 - 1}{2 - 1} = 3 \ imes \frac{32 - 1}{1} = 3 \ imes 31 = 93\n]", "This confirms that the sum of the first 5 terms in the geometric sequence starting with 3 and multiplying by 2 each time is 93.", "Understanding geometric sequences helps build a strong foundation in algebra and real-world problem solving. Whether calculating compound growth, modeling population growth, or analyzing investment returns, recognizing and working with geometric progressions is invaluable.", "Key takeaway:\nIn a geometric sequence starting with 3 and a common ratio of 2, the first 5 terms sum to 93, demonstrating the rapid growth possible with exponential multiplication."]
