Question: Find the fifth term and the nth term of the geometric sequence whose initial term \(a_1\) and…

Find the fifth term and the nth term of the geometric sequence whose initial term \(a_1\) and common ratio \(r\) are given.

\(a_1 = 7\), \(r = 2\)

Solution

To find the fifth term and the \(n\)th term of a geometric sequence, we use the formula for the \(n\)th term: \[ a_n = a_1 \cdot r^{n-1} \] The initial term \(a_1 = 7\) and the common ratio \(r = 2\). First, find the fifth term (\(a_5\)). Substitute 5 for \(n\): \[ a_5 = 7 \cdot 2^{5-1} \] Simplify the exponent: \[ a_5 = 7 \cdot 2^4 \] Calculate \(2^4\): \[ a_5 = 7 \cdot 16 \] Multiply: \[ a_5 = 112 \] Now, find the \(n\)th term (\(a_n\)). Use the general formula: \[ a_n = 7 \cdot 2^{n-1} \] The fifth term is 112 and the \(n\)th term is given by \(a_n = 7 \cdot 2^{n-1}\).