8=3 + 3+p÷4 + 3+2p÷4² and so on...find p

The equation you've provided appears to be recursive or possibly incomplete. I'll assume you want to solve for \( p \) in an expression of the form:

\[
8 = 3 + \frac{3 + p}{4} + \frac{3 + 2p}{4^2} + \cdots
\]

This resembles an infinite geometric series. Let's break it down.

### Step 1: Rewrite the Series
We can express the sum in terms of a geometric series:
\[
S = 3 + \frac{3 + p}{4} + \frac{3 + 2p}{16} + \cdots
\]

It appears the general term is of the form:
\[
\frac{3 + np}{4^n} \quad \text{for } n = 1, 2, 3, \dots
\]

### Step 2: Identify the Geometric Series Form
We aim to rewrite the infinite sum as a geometric series. To do this, we identify the first term (\(a\)) and the common ratio (\(r\)) of the series.

However, this series involves both constant terms and linear terms in \( p \), which makes the problem slightly more complex. Let’s first handle the terms separately.

### Step 3: Summing the Constant Terms
The constant terms form their own geometric series:
\[
S_{\text{constant}} = 3 + \frac{3}{4} + \frac{3}{16} + \cdots
\]
This is a geometric series with the first term \( a = 3 \) and common ratio \( r = \frac{1}{4} \). The sum of an infinite geometric series is given by:
\[
S_{\text{constant}} = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 4
\]

### Step 4: Summing the Terms Involving \( p \)
Now, consider the terms involving \( p \):
\[
S_p = \frac{p}{4} + \frac{2p}{16} + \cdots
\]
This is also a geometric series, with first term \( \frac{p}{4} \) and common ratio \( \frac{1}{4} \). The sum is:
\[
S_p = \frac{\frac{p}{4}}{1 - \frac{1}{4}} = \frac{\frac{p}{4}}{\frac{3}{4}} = \frac{p}{3}
\]

### Step 5: Add the Results Together
The total sum of the series is:
\[
S = S_{\text{constant}} + S_p = 4 + \frac{p}{3}
\]

### Step 6: Solve for \( p \)
We are given that the total sum equals 8:
\[
4 + \frac{p}{3} = 8
\]
Subtract 4 from both sides:
\[
\frac{p}{3} = 4
\]
Multiply both sides by 3:
\[
p = 12
\]

### Final Answer:
The value of \( p \) is \( \boxed{12} \).

Do you want any further details or clarifications? 

Here are 5 follow-up questions you might find useful:
1. How do you identify the terms of a geometric series?
2. What is the formula for summing an infinite geometric series?
3. How do constant and variable terms affect the overall sum of a series?
4. Can this method apply to finite sums as well?
5. How does changing the common ratio affect the series' convergence?

**Tip:** Always check if a series converges by confirming the common ratio is between -1 and 1.

8 = 3 + (3 + p)/4 + (3 + 2p)/4² and so on... find p

Learn how to solve for p in the infinite series equation: 8 = 3 + (3 + p)/4 + (3 + 2p)/4². This step-by-step solution covers geometric series concepts, algebraic manipulation, and the summation of infinite series. Suitable for students in Grades 10-12.

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