Problem 53 - Precalculus: Mathematics for Calculus

Question

Find the sum. $$\sum_{k=1}^{5} 2^{k-1}$$

Step-by-Step Solution

Answer

The sum is 31.

Step 1: Understand the Expression

The expression \( \sum_{k=1}^{5} 2^{k-1} \) represents the sum of the sequence terms from \( k = 1 \) to \( k = 5 \), where each term is given by the formula \( 2^{k-1} \).

Step 2: Write Out the Terms

Calculate each term individually:- For \( k = 1 \): \( 2^{1-1} = 2^{0} = 1 \)- For \( k = 2 \): \( 2^{2-1} = 2^{1} = 2 \)- For \( k = 3 \): \( 2^{3-1} = 2^{2} = 4 \)- For \( k = 4 \): \( 2^{4-1} = 2^{3} = 8 \)- For \( k = 5 \): \( 2^{5-1} = 2^{4} = 16 \)

Step 3: Compute the Sum

Add the individual terms together:\[ 1 + 2 + 4 + 8 + 16 = 31 \]

Step 4: Verify the Result

Check the calculations again to ensure the addition is correct. Each step reversely confirms that all terms have been calculated and added accurately resulting in \( 31 \).

Key Concepts

Summation Notation

Summation notation is a convenient way to express the addition of a series of numbers. It's often seen with a Greek capital sigma symbol (\( \Sigma \)), which stands for 'sum'. In a problem like \( \sum_{k=1}^{5} 2^{k-1} \), the index of summation \( k \) begins at 1 and goes up to 5, meaning we calculate the series for each integer value of \( k \) from 1 to 5.
This notation helps manage longer sums by providing a formula representing the series instead of listing every number. Each term to be summed is shown to the right of the symbol, helping mathematicians and students clearly see what calculations are necessary.
With summation, you can define a series succinctly. For example, in our expression, it tells us to calculate \( 2^{k-1} \) each time we increase \( k \) by 1, then add all the results. Summation notation can simplify and streamline solving sums, especially when dealing with sequences or patterns.

Exponents

Exponents provide a method to indicate repeated multiplication of the same number. In our exercise, the expression \( 2^{k-1} \) means the base number 2 is multiplied by itself \( k-1 \) times. This is a powerful way to express large computations succinctly.
Understanding exponents is key in geometric series, where each term is derived by raising a base number to a power defined by a sequence. For instance:

• For \( k = 1 \), \( 2^{1-1} = 2^{0} = 1 \)
• For \( k = 2 \), \( 2^{2-1} = 2^{1} = 2 \)
• For \( k = 3 \), \( 2^{3-1} = 2^{2} = 4 \)
• For \( k = 4 \), \( 2^{4-1} = 2^{3} = 8 \)
• For \( k = 5 \), \( 2^{5-1} = 2^{4} = 16 \)

Each term in the geometric series increases by a common factor, here the factor of 2. Learning how exponents work allows one to understand how to generate sequence terms efficiently and predict later terms without direct multiplication.

Sequence Terms

In mathematics, sequence terms are individual elements in a sequence based on a specific rule or formula. For the exercise \( \sum_{k=1}^{5} 2^{k-1} \), each term is derived by substituting \( k \) in the expression \( 2^{k-1} \). You get five terms since \( k \) ranges from 1 through 5. These terms are directly calculated as follows:

• The first term (when \( k=1 \)) is \( 1 \).
• The second term (when \( k=2 \)) is \( 2 \).
• The third term (when \( k=3 \)) is \( 4 \).
• The fourth term (when \( k=4 \)) is \( 8 \).
• The fifth term (when \( k=5 \)) is \( 16 \).

Adding these sequence terms gives the sum \( 31 \). By understanding how sequence terms are structured using formulas, especially in geometric series, you can quickly compute sums and understand patterns in various mathematical sequences. Sequence terms creation forms the foundation for solving many problems involving series and sequences.