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Chapter 14: Problem 33

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

Short Answer

Expert verified
The series can be represented in summation notation as \(\sum_{i=1}^{11} 2^{i}\).

Step by step solution

01

Identifying the sequence

The given series is a geometric sequence with the common ratio of 2. This is evident as each term of the sequence is obtained by multiplying the preceding term by 2.
02

Deciding the index of summation

Since the problem specifies to start the lower limit of summation at 1 and use i as the index of summation, the first term should be \(2^{1}=2\), the second term \(2^{2}=4\), and so on. You can see that the ith term of the sequence is \(2^{i}\).
03

Writing in Sigma Notation

The Sigma notation for a geometric series follows the format \(\sum_{i=1}^{n} ar^{i-1}\). Here, n is the number of terms a is the first term and r is the common ratio. Substituting the values in the formula, the series can be represented as \(\sum_{i=1}^{11} 2^{i}\).

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