Evaluate the Summation sum from k=1 to 7 of 36-2^k
The problem involves finding the sum of a series, where each term of the series is given by a specific mathematical expression: 36 minus 2 raised to the power of k. The variable k is the index of summation. It starts at 1 and progresses through integer increments until k equals 7. To evaluate the summation, one would typically calculate the value of the expression for each integer value of k from 1 to 7, and then add all these values together to find the total sum.
$\sum_{k = 1}^{7} 36 - 2^{k}$
Answer
Solution:
Step 1: Decompose the given summation into two separate summations.
$\sum_{k = 1}^{7} (36 - 2^{k}) = \sum_{k = 1}^{7} 36 - \sum_{k = 1}^{7} 2^{k}$
Step 2: Calculate the summation of the constant term.
Step 2.1: Use the summation formula for a constant.
$\sum_{k = 1}^{n} c = cn$
Step 2.2: Plug in the values for the constant summation.
$(36)(7)$
Step 2.3: Perform the multiplication.
$252$
Step 3: Calculate the summation of the geometric series.
Step 3.1: Apply the formula for the sum of a geometric series.
$S = a \left(\frac{1 - r^{n}}{1 - r}\right)$
Step 3.2: Determine the common ratio of the series.
Step 3.2.1: Insert the terms into the ratio formula.
$r = \frac{-2^{k+1}}{-2^{k}}$
Step 3.2.2: Simplify the ratio.
Step 3.2.2.1: Recognize that dividing two negatives yields a positive.
$r = \frac{2^{k+1}}{2^{k}}$
Step 3.2.2.2: Reduce the common factors.
Step 3.2.2.2.1: Factor out $2^{k}$ from $2^{k+1}$.
$r = \frac{2^{k} \cdot 2}{2^{k}}$
Step 3.2.2.2.2: Simplify by canceling out common terms.
$r = \frac{2}{1}$
Step 3.2.2.2.3: Finalize the common ratio.
$r = 2$
Step 3.3: Identify the first term of the series.
Step 3.3.1: Substitute $k=1$ into the term $-2^{k}$.
$a = -2^{1}$
Step 3.3.2: Simplify to find the first term.
$a = -2$
Step 3.4: Insert the values into the geometric series sum formula.
$S = -2 \left(\frac{1 - 2^{7}}{1 - 2}\right)$
Step 3.5: Simplify the expression.
Step 3.5.1: Work on the numerator.
Step 3.5.1.1: Calculate $2^{7}$.
$S = -2 \left(\frac{1 - 128}{1 - 2}\right)$
Step 3.5.1.2: Perform the subtraction in the numerator.
$S = -2 \left(\frac{-127}{1 - 2}\right)$
Step 3.5.2: Simplify the denominator.
Step 3.5.2.1: Subtract $2$ from $1$.
$S = -2 \left(\frac{-127}{-1}\right)$
Step 3.5.3: Divide the numerator by the denominator.
$S = -2 \cdot 127$
Step 3.5.4: Multiply to find the sum of the geometric series.
$S = -254$
Step 4: Combine the results of the two summations.
$252 - 254$
Step 5: Final calculation.
$-2$
Knowledge Notes:
The problem involves evaluating a summation that consists of a constant term and a geometric series. The solution process includes several key knowledge points:
Summation of a constant: The sum of a constant $c$ over $n$ terms is simply $cn$.
Geometric series: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The sum of the first $n$ terms of a geometric series can be calculated using the formula $S = a \left(\frac{1 - r^{n}}{1 - r}\right)$, where $a$ is the first term.
Simplification of ratios: When simplifying the ratio of two terms in a geometric series, any common factors in the numerator and denominator can be canceled out.
Algebraic manipulation: The solution involves basic algebraic manipulations such as multiplication, division, and simplification of expressions.
Combining summations: When dealing with a summation of multiple terms, it is often helpful to separate the summation into individual components, solve each one, and then combine the results.
