(d) $$ \frac{6^{n+2}+7 imes 6^{n}}{6^{n+1} imes 8-5 imes 6^{n}}=1 $$

Question

(d) $$ \frac{6^{n+2}+7 	imes 6^{n}}{6^{n+1} 	imes 8-5 	imes 6^{n}}=1 $$

UpStudy AI · Accepted Answer

1. Write the given equation:
   $$
   \frac{6^{n+2} + 7 \times 6^n}{8 \times 6^{n+1} - 5 \times 6^n} = 1
   $$

2. Factor $$ 6^n $$ from the numerator and the denominator:
   $$
   \text{Numerator: } 6^{n+2} + 7 \times 6^n = 6^n \left(6^2 + 7\right) = 6^n \left(36 + 7\right) = 6^n \times 43
   $$
   $$
   \text{Denominator: } 8 \times 6^{n+1} - 5 \times 6^n = 6^n \left(8 \times 6 - 5\right) = 6^n \left(48 - 5\right) = 6^n \times 43
   $$

3. Substitute the factored forms back into the equation:
   $$
   \frac{6^n \times 43}{6^n \times 43} = 1
   $$

4. Since $$ 6^n \times 43 $$ is nonzero for all values of $$ n $$ (because $$ 6^n > 0 $$ for any real $$ n $$), the expression simplifies to:
   $$
   1 = 1
   $$

5. The equality holds for all $$ n $$.

Thus, the solution is that the equation is true for all values of $$ n $$.
The equation is true for all values of $$ n $$.

(d)

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