Evaluate the summation from n=1 to 11 of (3/4)^n.

Steps to Solve

1. Identify the geometric series parameters The summation given is from $n=1$ to $n=11$ of the expression $\left(\frac{3}{4}\right)^n$. This is a geometric series where:

   • The first term ( a = \left(\frac{3}{4}\right)^1 = \frac{3}{4} )
   • The common ratio ( r = \frac{3}{4} )
   • The number of terms ( n = 11 )

2. Apply the formula for the sum of a geometric series The formula for the sum ( S_n ) of the first ( n ) terms of a geometric series is: $$ S_n = a \frac{1 - r^n}{1 - r} $$ Substituting the identified values:

   • ( a = \frac{3}{4} )
   • ( r = \frac{3}{4} )
   • ( n = 11 )

So we have: $$ S_{11} = \frac{3}{4} \frac{1 - \left(\frac{3}{4}\right)^{11}}{1 - \frac{3}{4}} $$

3. Calculate the denominator The denominator simplifies as follows: $$ 1 - \frac{3}{4} = \frac{1}{4} $$

4. Substitute and simplify the sum Now substituting back into the formula: $$ S_{11} = \frac{3}{4} \frac{1 - \left(\frac{3}{4}\right)^{11}}{\frac{1}{4}} $$ This simplifies to: $$ S_{11} = 3 \left(1 - \left(\frac{3}{4}\right)^{11}\right) $$

5. Calculate (\left(\frac{3}{4}\right)^{11}) We need to compute: $$ \left(\frac{3}{4}\right)^{11} \approx 0.031676352 $$ Now plug this back into the equation: $$ S_{11} = 3 \left(1 - 0.031676352\right) $$

6. Final calculation Compute the final sum: $$ S_{11} = 3 \times (0.968323648) \approx 2.904970944 $$