Questions: The formula for the nth term an of a sequence an is given below. Find the values of a1, a2, a3, and a4 an = 4^n / 4^(n+2) a1 =

Transcript text: The formula for the $n$th term $a_{n}$ of a sequence $\left\{a_{n}\right\}$ is given below. Find the values of $a_{1}, a_{2}, a_{3}$, and $a_{4}$ \[ a_{n}=\frac{4^{n}}{4^{n+2}} \] $a_{1}=\square$

Solution

Solution Steps

To find the values of $ a_1, a_2, a_3, $ and $ a_4 $ for the given sequence, we need to substitute $ n = 1, 2, 3, $ and $ 4 $ into the formula $ a_n = \frac{4^n}{4^{n+2}} $. Simplify the expression for each value of $ n $ to find the corresponding term in the sequence.

Step 1: Calculate $ a_1 $

To find $ a_1 $, we substitute $ n = 1 $ into the formula: \[ a_1 = \frac{4^1}{4^{1+2}} = \frac{4}{4^3} = \frac{4}{64} = 0.0625 \]

Step 2: Calculate $ a_2 $

Next, we calculate $ a_2 $ by substituting $ n = 2 $: \[ a_2 = \frac{4^2}{4^{2+2}} = \frac{16}{4^4} = \frac{16}{256} = 0.0625 \]

Step 3: Calculate $ a_3 $

Now, we find $ a_3 $ by substituting $ n = 3 $: \[ a_3 = \frac{4^3}{4^{3+2}} = \frac{64}{4^5} = \frac{64}{1024} = 0.0625 \]

Step 4: Calculate $ a_4 $

Finally, we calculate $ a_4 $ by substituting $ n = 4 $: \[ a_4 = \frac{4^4}{4^{4+2}} = \frac{256}{4^6} = \frac{256}{4096} = 0.0625 \]

Final Answer

The values of the sequence are: \[ a_1 = 0.0625, \quad a_2 = 0.0625, \quad a_3 = 0.0625, \quad a_4 = 0.0625 \] Thus, the final boxed answer is: \[ \boxed{a_1 = 0.0625, \, a_2 = 0.0625, \, a_3 = 0.0625, \, a_4 = 0.0625} \]

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