Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $2+2^2+2^3+\cdots +2^{11}$

Verified step by step guidance

1. Identify the pattern in the sum: the terms are powers of 2 starting from 2^1 up to 2^{11}.
2. Recognize that the index of summation, i, will represent the exponent on 2 in each term.
3. Set the lower limit of summation to 1, since the first term is 2^1.
4. Set the upper limit of summation to 11, since the last term is 2^{11}.
5. Write the summation notation as \(\sum_{i=1}^{11} 2^{i}\) to represent the entire sum.

Key Concepts

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. For example, ∑ from i=1 to n of a_i represents adding terms a_1 through a_n.

Exponents and Powers

Exponents indicate repeated multiplication of a base number. For instance, 2^3 means 2 multiplied by itself three times (2×2×2=8). Understanding how to express terms with exponents is essential when writing sums involving powers in summation notation.

Index of Summation and Limits

The index of summation (commonly i) represents the variable that changes in each term of the sum. The lower limit is the starting value of i, and the upper limit is the ending value. Correctly setting these limits ensures the sum includes all intended terms.