We are tasked with solving for the first term $a$ and the common ratio $r$ of a geometric series, given the following information:

1. The sum of the series without the first term is $-4$.
2. The sum of the series starting from the second term is $21$.

Let’s break this down step by step.

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General Formula for the Sum of an Infinite Series

The sum of an infinite geometric series is given by:

$S = \frac{a}{1 - r}, \quad \text{where } |r| < 1$

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Information 1: Sum of the Series Without the First Term

If the first term $a$ is excluded, the remaining series begins with $ar$. The sum of this remaining series is:

$S_{\text{without first term}} = \frac{ar}{1 - r}$

We are given:

\frac{ar}{1 - r} = -4 \tag{1}

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Information 2: Sum of the Series with the Second Term

If the series starts from the second term, the first term is $ar$ instead of $a$, and the sum is:

$S_{\text{starting from second}} = \frac{ar}{1 - r}$

We are given:

\frac{a}{1 - r} = 21 \tag{2}

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Step 1: Solving the System of Equations

From Equation (2):

a = 21(1 - r) \tag{3}

Substitute $a$ from Equation (3) into Equation (1):

$\frac{21(1 - r)r}{1 - r} = -4$

Simplify:

$21r = -4$

r = -\frac{4}{21} \tag{4}

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Step 2: Find $a$

Substitute $r = -\frac{4}{21}$ into Equation (3):

$a = 21 \left( 1 - \left(-\frac{4}{21}\right) \right)$

$a = 21 \left( 1 + \frac{4}{21} \right)$

$a = 21 \cdot \frac{25}{21}$

a = 25 \tag{5}

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Final Answer

1. First term ($a$): $25$
2. Common ratio ($r$): $-\frac{4}{21}$

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Would you like a detailed verification or further clarification?

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Related Questions

1. How is the sum of an infinite series derived mathematically?
2. Why must $|r| < 1$ for the sum of a geometric series to converge?
3. How does excluding the first term change the formula of the series?
4. Can a geometric series converge to a positive value if $r < 0$?
5. How can this problem be adapted to include finite series?

Tip

Always verify your results by substituting them back into the original equations to ensure consistency.