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

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 \( n \).