∑ (from n=1 to 11) (3/4)^n

Steps to Solve

1. Identify the summation formula

The summation can be expressed as:

$$ S = \sum_{n=1}^{11} \left(\frac{3}{4}\right)^n $$

2. Use the formula for a geometric series

The sum of a geometric series can be calculated using the formula:

$$ S_n = a \frac{1 - r^n}{1 - r} $$

where:

• ( S_n ) is the sum of the series
• ( a ) is the first term
• ( r ) is the common ratio
• ( n ) is the number of terms

In this case, ( a = \frac{3}{4} ), ( r = \frac{3}{4} ), and ( n = 11 ).

3. Substitute values into the formula

Now, substitute ( a ), ( r ), and ( n ) into the formula:

$$ S = \left(\frac{3}{4}\right) \frac{1 - \left(\frac{3}{4}\right)^{11}}{1 - \frac{3}{4}} $$

4. Simplify the denominator

Calculate the denominator:

$$ 1 - \frac{3}{4} = \frac{1}{4} $$

5. Substitute and simplify further

Replace the denominator in the sum:

$$ S = \left(\frac{3}{4}\right) \frac{1 - \left(\frac{3}{4}\right)^{11}}{\frac{1}{4}} $$

This simplifies to:

$$ S = 3 \left(1 - \left(\frac{3}{4}\right)^{11}\right) $$

6. Calculate ( \left(\frac{3}{4}\right)^{11} )

Calculate ( \left(\frac{3}{4}\right)^{11} ):

Using a calculator,

$$ \left(\frac{3}{4}\right)^{11} \approx 0.031676352 $$

7. Final substitution and computation

Now substitute back into the equation:

$$ S \approx 3 \left(1 - 0.031676352\right) \approx 3 (0.968323648) $$

This gives us:

$$ S \approx 2.904970944 $$