4) $$ \frac{a}{1-r}-\frac{a\left(r^{n}-1\right)}{r-1}

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4) $$ \frac{a}{1-r}-\frac{a\left(r^{n}-1\right)}{r-1}<1 $$

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Let's solve the inequality: $$ \frac{a}{1 - r} - \frac{a(r^{n} - 1)}{r - 1} < 1 $$ ### Step 1: Simplify the Expression First, observe that $$ r - 1 = - (1 - r) $$. This allows us to rewrite the second term: $$ \frac{a(r^{n} - 1)}{r - 1} = \frac{a(r^{n} - 1)}{-(1 - r)} = -\frac{a(r^{n} - 1)}{1 - r} $$ Now, substitute this back into the original inequality: $$ \frac{a}{1 - r} - \left(-\frac{a(r^{n} - 1)}{1 - r}\right) < 1 $$ $$ \frac{a}{1 - r} + \frac{a(r^{n} - 1)}{1 - r} < 1 $$ Combine the fractions: $$ \frac{a + a(r^{n} - 1)}{1 - r} < 1 $$ $$ \frac{a \cdot r^{n}}{1 - r} < 1 $$ ### Step 2: Solve for $$ a $$ The simplified inequality is: $$ \frac{a \cdot r^{n}}{1 - r} < 1 $$ Let's solve for $$ a $$: $$ a \cdot r^{n} < 1 - r $$ $$ a < \frac{1 - r}{r^{n}} $$ ### Step 3: Consider the Sign of $$ 1 - r $$ The inequality depends on the value of $$ r $$. We'll analyze two primary cases: 1. **When $$ 0 < r < 1 $$:** - $$ 1 - r > 0 $$ and $$ r^{n} > 0 $$. - The inequality simplifies to: $$ a < \frac{1 - r}{r^{n}} $$ - **Conclusion:** For the inequality to hold, $$ a $$ must be **less than** $$ \frac{1 - r}{r^{n}} $$. 2. **When $$ r > 1 $$:** - $$ 1 - r < 0 $$ and $$ r^{n} > 0 $$. - Multiplying both sides by $$ 1 - r $$ (which is negative) reverses the inequality: $$ a \cdot r^{n} > 1 - r $$ - Since $$ 1 - r < 0 $$ and $$ r^{n} > 0 $$, $$ a $$ only needs to be greater than a negative number, which is always true if $$ a > 0 $$. - **Conclusion:** For $$ r > 1 $$, **any positive $$ a $$** satisfies the inequality. 3. **When $$ r = 1 $$:** - The original expression is undefined due to division by zero. 4. **When $$ r < 0 $$:** - Additional considerations are needed based on the parity of $$ n $$ and the specific context, but generally, the above conclusions hold for $$ |r| > 0 $$. ### Final Summary - **If $$ 0 < r < 1 $$:** $$ a < \frac{1 - r}{r^{n}} $$ - **If $$ r > 1 $$ and $$ a > 0 $$:** The inequality is always satisfied. - **If $$ r = 1 $$:** The inequality is undefined. - **If $$ r < 0 $$:** Further analysis is required based on the specific values of $$ r $$ and $$ n $$. To solve the inequality $$ \frac{a}{1 - r} - \frac{a(r^{n} - 1)}{r - 1} < 1 $$, follow these steps: 1. **Simplify the Expression:** $$ \frac{a}{1 - r} + \frac{a(r^{n} - 1)}{1 - r} < 1 $$ $$ \frac{a \cdot r^{n}}{1 - r} < 1 $$ 2. **Solve for $$ a $$:** $$ a < \frac{1 - r}{r^{n}} $$ 3. **Consider the Value of $$ r $$:** - **For $$ 0 < r < 1 $$:** $$ a < \frac{1 - r}{r^{n}} $$ - **For $$ r > 1 $$ and $$ a > 0 $$:** The inequality always holds. - **For $$ r = 1 $$ or $$ r < 0 $$:** The inequality is undefined or requires further analysis. **Conclusion:** The value of $$ a $$ must be less than $$ \frac{1 - r}{r^{n}} $$ when $$ 0 < r < 1 $$. For $$ r > 1 $$ and $$ a > 0 $$, the inequality is always satisfied.

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