Question

Evaluate the sums. \(\sum_{i=1}^{4} 2^{i}\)

Short answer

Answer: The value of the sum is 30.

Step-by-step solution

Write out each term in the sum

The sum can be written as: \(2^1 + 2^2 + 2^3 + 2^4\)

Calculate the values of each term

Find the value of each term: \(2^1 = 2\) \(2^2 = 4\) \(2^3 = 8\) \(2^4 = 16\)

Add the values together

Now, add the calculated values together: \(2 + 4 + 8 + 16 = 30\) So, the value of the sum \(\sum_{i=1}^{4} 2^{i}\) is 30.

Key concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are written as a superscript number to the right of the base number. For instance, in the expression \(2^4\), 2 is the base and 4 is the exponent. This means 2 is multiplied by itself 4 times, or \(2 \times 2 \times 2 \times 2 = 16\).

• An exponent of 1 means the number remains unchanged. For example, \(2^1 = 2\).
• An exponent of 0 means the result is always 1, regardless of the base, as shown in \(2^0 = 1\).

Understanding exponents is crucial when working with powers and performing operations that involve repeated multiplication, as seen in arithmetic sequences and summation problems. They simplify expressions and calculations significantly.

Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant to the previous term. This constant is known as the "common difference." Let’s consider a simple example: the sequence 2, 4, 6, 8 is an arithmetic sequence with a common difference of 2.

In the context of the given exercise, though it deals with geometric progression (specifically with powers of 2), the concept of step-by-step addition is similar to working through an arithmetic sequence. Each calculated power of 2 is like a term that needs to be evaluated and then summed sequentially. This systematic approach of finding terms and later summing them can help in thoroughly understanding how sequences and series operate in mathematics.

Mathematical Notation

Mathematical notation uses symbols to concisely and clearly express mathematical concepts. In the exercise, the summation notation \(\sum\) represents the sum of a sequence of numbers, identified here as powers of 2, from 1 to 4.

• The notation \(\sum_{i=1}^{4} 2^{i}\) tells us to sum the expression \(2^i\) as the variable \(i\) goes from 1 to 4.
• \(i=1\) is the starting point, also known as the lower limit of the summation.
• \(i=4\) is the ending point, or the upper limit of the summation.

This type of notation is powerful as it allows for the concise expression of what might otherwise require a long series of terms to write out. Understanding and practicing mathematical notation, like summation, aids in effectively communicating and solving mathematical problems, making it a fundamental tool in math, especially in sequences and series.