In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 2+2^2+2^3+⋯+ 2^11

Verified step by step guidance

1. Identify the pattern in the series: The given series is 2 + 2^2 + 2^3 + ... + 2^{11}.
2. Recognize that each term in the series can be expressed as 2^i, where i is the index of summation.
3. Determine the range of the index: The series starts at 2^1 and ends at 2^{11}, so the index i ranges from 1 to 11.
4. Write the series in summation notation: Use the summation symbol \( \sum \) to express the series as \( \sum_{i=1}^{11} 2^i \).
5. Verify that the summation notation correctly represents the original series by checking the lower and upper limits and the expression for each term.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It typically uses the Greek letter sigma (Σ) to denote the sum, with limits indicating the starting and ending indices. For example, Σ from i=1 to n of a_i represents the sum of the terms a_1, a_2, ..., a_n.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In the context of the given question, the terms 2^1, 2^2, ..., 2^11 represent an exponential sequence where the base is 2, and the exponent increases incrementally.

Index of Summation

The index of summation is a variable used to represent the position of terms in a sequence being summed. In this case, 'i' serves as the index, starting from 1 and increasing by 1 for each term until it reaches 11. This allows for a concise representation of the sum of terms in a series.