Final answer:
The sum of the provided geometric sequence 1, 2, 4, 8, 16 is 31, obtained by adding all the terms together.
Step-by-step explanation:
The sum of the geometric sequence 1, 2, 4, 8, 16 is 31.Geometric sequences expand by a consistent factor with each term. In this sequence, each term is double the previous one. To find the sum of a geometric sequence, sum the terms: 1 + 2 + 4 + 8 + 16 = 31. This particular sequence is a finite geometric series, where the ratio (common multiplier) is 2. Start with the first term and continue multiplying by 2 until the last term is reached, summing all the terms.The sum of a geometric sequence can be found using the formula:
Sum = a * (1 - r^n) / (1 - r)where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.In this case, the first term is 1 and the common ratio is 2. The sum can be calculated as:Sum = 1 * (1 - 2^n) / (1 2)Simplifying the expression gives:Sum = (2^n - 1) / (1 - 2)Since we are asked to find the sum of the sequence up to 'a', we need to find the value of 'n' that corresponds to the given 'a'.Solving the equation 2^n = a, we find that n = log(base 2) of a.Plugging in the value of 'a' into the formula:Sum = (2^(log(base 2) of a) - 1) / (1 - 2)Sum = a - 1Therefore, the sum of the geometric sequence is a - 1.