Questions: If the first 4 terms of a geometric sequence are 5,20,80,320, then the formula for the nth term in the sequence is a.) an=4 * 5^(n-1) b.) an=5 (n-1)^4 c.) an=5 4^(n-1) d.) an=4 *(n-1)^5

If the first 4 terms of a geometric sequence are 5,20,80,320, then the formula for the nth term in the sequence is
a.) an=4 * 5^(n-1)
b.) an=5 *(n-1)^4
c.) an=5 * 4^(n-1)
d.) an=4 *(n-1)^5

Transcript text: If the first 4 terms of a geometric sequence are $\{5,20,80,320\}$, then the formula for the $n^{\text {th }}$ term in the sequence is $\qquad$ a.) $a_{n}=4 \cdot 5^{n-1}$ b.) $a_{n}=5 \cdot(n-1)^{4}$ c.) $a_{n}=5 \cdot 4^{n-1}$ d.) $a_{n}=4 \cdot(n-1)^{5}$

Solution

Solution Steps

To find the formula for the $ n^{\text{th}} $ term of a geometric sequence, we need to identify the first term and the common ratio. The first term $ a_1 $ is given as 5. The common ratio $ r $ can be found by dividing the second term by the first term, the third term by the second term, and so on. Once we have the common ratio, we can use the general formula for the $ n^{\text{th}} $ term of a geometric sequence, which is $ a_n = a_1 \cdot r^{n-1} $.

Step 1: Identify the First Term and Common Ratio

The first term of the geometric sequence is given as $ a_1 = 5 $. To find the common ratio $ r $, we divide the second term by the first term: \[ r = \frac{20}{5} = 4 \]

Step 2: General Formula for the $ n^{\text{th}} $ Term

The general formula for the $ n^{\text{th}} $ term of a geometric sequence is: \[ a_n = a_1 \cdot r^{n-1} \] Substituting $ a_1 = 5 $ and $ r = 4 $: \[ a_n = 5 \cdot 4^{n-1} \]

Step 3: Match the Formula with Given Options

We compare the derived formula $ a_n = 5 \cdot 4^{n-1} $ with the given options: