To solve the problem, we can use the following formulas related to an infinite geometric series:

1. Sum of an infinite geometric series:
   $S_{\infty} = \frac{a}{1 - q}$ where $a$ is the first term and $q$ is the common ratio ($-1 < q < 1$).

2. Sum of the first $n$ terms of a geometric series:
   $S_n = a \frac{1 - q^n}{1 - q}$

Step 1: Use $S_{\infty} = 16$

From the sum of the infinite series: \frac{a}{1 - q} = 16 \tag{1}

Step 2: Use $S_4 = 15$

From the sum of the first four terms: a \frac{1 - q^4}{1 - q} = 15 \tag{2}

Step 3: Solve for $a$ and $q$

From Equation (1): a = 16(1 - q) \tag{3}

Substitute $a = 16(1 - q)$ into Equation (2): $16(1 - q) \frac{1 - q^4}{1 - q} = 15$

Simplify by canceling $1 - q$ (since $q \neq 1$): $16(1 - q^4) = 15$

Expand $1 - q^4$ using the difference of squares: $1 - q^4 = (1 - q^2)(1 + q^2)$

Thus: 16(1 - q^2)(1 + q^2) = 15 \tag{4}

Divide through by 16: $(1 - q^2)(1 + q^2) = \frac{15}{16}$

Step 4: Solve for $q$

Let $q^2 = x$ (substitute $x$ for simplicity): $(1 - x)(1 + x) = \frac{15}{16}$

Simplify: $1 - x^2 = \frac{15}{16}$

$x^2 = 1 - \frac{15}{16} = \frac{1}{16}$

So: $x = \sqrt{\frac{1}{16}} = \frac{1}{4}$

Thus: $q^2 = \frac{1}{4} \quad \implies \quad q = \pm \frac{1}{2}$

Step 5: Find $a$

Substitute $q = \frac{1}{2}$ into Equation (3): $a = 16(1 - \frac{1}{2}) = 16 \cdot \frac{1}{2} = 8$

Final Answer:

• First term ($a$): $8$
• Common ratio ($q$): $\frac{1}{2}$

Verification:

1. Sum of infinite series: $S_{\infty} = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16$
2. Sum of the first 4 terms: $S_4 = 8 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 8 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 8 \frac{\frac{15}{16}}{\frac{1}{2}} = 8 \cdot \frac{15}{8} = 15$

Both conditions are satisfied.

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Would you like a detailed explanation of any step? Here are five related questions to explore further:

1. What happens if $q = -\frac{1}{2}$ instead of $\frac{1}{2}$?
2. How does the sum of the series depend on the common ratio $q$?
3. What changes in the formulas if $q$ is not between $-1$ and $1$?
4. How do we calculate the sum for a different number of terms, such as $S_6$?
5. Can we graph the terms of the series to visualize the convergence?

Tip: Always verify results by substituting them back into the original problem equations!