Problem

Find any of the values $a_{1}, r, a_{n}, n$, or $S_{n}$ that are missing from the geometric sequence. \[ r=\frac{5}{3}, n=5, S_{5}=\frac{1441}{3} \] The value of $a_{1}$ is 27 . (Simplify your answer. Type an integer or a fraction.) The value of $a_{5}$ is (Simplify your answer. Type an integer or a fraction.)

Solution

Step 1 :We are given the common ratio \(r\), the number of terms \(n\), the sum of the first \(n\) terms \(S_{n}\), and the first term \(a_{1}\). We are asked to find the fifth term \(a_{5}\).

Step 2 :The formula for the nth term of a geometric sequence is \(a_{n} = a_{1} * r^{(n-1)}\). We can use this formula to find \(a_{5}\).

Step 3 :Substitute the given values into the formula: \(a_{1} = 27\), \(r = \frac{5}{3}\), and \(n = 5\).

Step 4 :Calculate \(a_{5} = 27 * (\frac{5}{3})^{(5-1)} = \frac{625}{3}\).

Step 5 :Final Answer: The value of \(a_{5}\) is \(\boxed{\frac{625}{3}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/o87Zgd3KJk/

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