In each of the following geometric sequences, find (i) the 5 th term; (ii) the nth term.
a.

Ask by Li Deleon. in the United States

Nov 27,2024

Question

In each of the following geometric sequences, find (i) the 5 th term; (ii) the nth term. a. $$ \frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \ldots $$

UpStudy AI · Accepted Answer

To find the 5th term and the nth term of the given geometric sequence $$ \frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \ldots $$, we first need to identify the first term and the common ratio.

1. **Identify the first term and common ratio:**
   - The first term $$ a = \frac{1}{2} $$.
   - To find the common ratio $$ r $$, we can divide the second term by the first term:
     $$
     r = \frac{\frac{3}{4}}{\frac{1}{2}} = \frac{3}{4} \times \frac{2}{1} = \frac{3 \times 2}{4 \times 1} = \frac{6}{4} = \frac{3}{2}.
     $$

2. **General formula for the nth term:**
   The nth term of a geometric sequence can be expressed as:
   $$
   a_n = a \cdot r^{n-1}.
   $$
   Substituting the values of $$ a $$ and $$ r $$:
   $$
   a_n = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{n-1}.
   $$

3. **Find the 5th term:**
   To find the 5th term $$ a_5 $$:
   $$
   a_5 = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{5-1} = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{4}.
   $$
   Now calculate $$ \left(\frac{3}{2}\right)^{4} $$:
   $$
   \left(\frac{3}{2}\right)^{4} = \frac{3^4}{2^4} = \frac{81}{16}.
   $$
   Therefore,
   $$
   a_5 = \frac{1}{2} \cdot \frac{81}{16} = \frac{81}{32}.
   $$

4. **Final results:**
   - The 5th term $$ a_5 = \frac{81}{32} $$.
   - The nth term $$ a_n = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{n-1} $$.

Thus, the answers are:
(i) The 5th term is $$ \frac{81}{32} $$.  
(ii) The nth term is $$ a_n = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{n-1} $$.
(i) The 5th term is $$ \frac{81}{32} $$. (ii) The nth term is $$ a_n = \frac{1}{2} \cdot \left(\frac{3}{2}\right)^{n-1} $$.

In each of the following geometric sequences, find (i) the 5 th term; (ii) the nth term.
a.

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