AI tutor
Welcome to Bytelearn!
Let’s check out your problem:
The common ratio of a geometric series is and the sum of the first terms is .What is the first term of the series?
Full solution
Q. The common ratio of a geometric series is and the sum of the first terms is .What is the first term of the series?
- Geometric series sum formula: The sum of the first terms of a geometric series is given by the formula:where is the sum of the first terms, is the first term, is the common ratio, and is the number of terms.
- Given values: We are given the sum of the first terms is , the common ratio is , and we need to find the first term .
- Substitute values into formula: Substitute the given values into the sum formula:
- Simplify the equation: Simplify the equation:
- Isolate the term involving : Multiply both sides by to isolate the term involving :
- Calculate left side of the equation: Calculate the left side of the equation: $\(170\) \times \left(\frac{\(3\)}{\(4\)}\right) = \(127\).\(5\)
- Equation with simplified right side: Now we have: \(127.5 = a_1(1 - \frac{1}{256})\)
- Divide both sides to solve for \(a_1\): Simplify the right side of the equation: \(127.5 = a_1\left(\frac{255}{256}\right)\)
- Calculate the value of \(a_1\): Divide both sides by \((255/256)\) to solve for \(a_1\):\(\newline\)\[a_1 = \frac{127.5}{(255/256)}\]
- Perform the multiplication: Calculate the value of \(a_1\): \(\newline\)\[a_1 = 127.5 \times \left(\frac{256}{255}\right)\]
- Perform the multiplication: Calculate the value of \(a_1\): \(\newline\)\[a_1 = 127.5 \times \left(\frac{256}{255}\right)\]Perform the multiplication: \(\newline\)\[a_1 = 128\]
