If $$ \sum_{k=p}^{\infty} 4.3^{2-k}=\frac{2}{9} $$, determine the value of $$ p $$

Question

UpStudy AI · Accepted Answer

To determine the value of $$ p $$ in the equation:

$$
\sum_{k=p}^{\infty} 4 \cdot 3^{2-k} = \frac{2}{9},
$$

let's break it down step by step.

### Step 1: Simplify the Series

First, express $$ 3^{2-k} $$ as:

$$
3^{2-k} = 3^2 \cdot 3^{-k} = 9 \cdot 3^{-k}.
$$

Substituting this into the series:

$$
\sum_{k=p}^{\infty} 4 \cdot 3^{2-k} = \sum_{k=p}^{\infty} 4 \cdot 9 \cdot 3^{-k} = \sum_{k=p}^{\infty} 36 \cdot 3^{-k}.
$$

### Step 2: Recognize the Geometric Series

The series:

$$
\sum_{k=p}^{\infty} 36 \cdot 3^{-k}
$$

is a geometric series with the first term $$ a = 36 \cdot 3^{-p} $$ and common ratio $$ r = \frac{1}{3} $$.

The sum of an infinite geometric series is given by:

$$
S = \frac{a}{1 - r}.
$$

Substituting the values:

$$
S = \frac{36 \cdot 3^{-p}}{1 - \frac{1}{3}} = \frac{36 \cdot 3^{-p}}{\frac{2}{3}} = 54 \cdot 3^{-p}.
$$

### Step 3: Set Up the Equation

According to the problem:

$$
54 \cdot 3^{-p} = \frac{2}{9}.
$$

Multiply both sides by $$ 3^p $$:

$$
54 = \frac{2}{9} \cdot 3^p.
$$

Multiply both sides by $$ \frac{9}{2} $$:

$$
54 \cdot \frac{9}{2} = 3^p \implies 243 = 3^p.
$$

### Step 4: Solve for $$ p $$

Since $$ 243 = 3^5 $$, it follows that:

$$
p = 5.
$$

**Answer:** $$ \boxed{5} $$
$$ p = 5 $$

If \( \sum_{k=p}^{\infty} 4.3^{2-k}=\frac{2}{9} \), determine the value of \( p \)