In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+ar+ar2+⋯+ ar^12

Verified step by step guidance

1. <Identify the pattern in the series: The given series is a geometric series with the first term 'a' and common ratio 'r'.>
2. <Recognize the general form of a geometric series: The general term of a geometric series can be expressed as ar^k, where k is the index of summation.>
3. <Determine the range of the index: The series starts with ar^0 and ends with ar^12, so the index k will range from 0 to 12.>
4. <Write the series in summation notation: Use the summation symbol \( \sum \) to express the series.>
5. <Express the series as \( \sum_{k=0}^{12} ar^k \), where k is the index of summation.>

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It typically involves the Greek letter sigma (Σ) and includes an index of summation, a lower limit, and an upper limit. For example, Σ from k=0 to n of a_k indicates the sum of the terms a_k as k varies from 0 to n.

Geometric Series

A geometric series is a series of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The series a + ar + ar^2 + ... + ar^n can be expressed in summation notation, where 'a' is the first term and 'r' is the common ratio. Understanding the properties of geometric series is essential for correctly expressing them in summation form.

Index of Summation

The index of summation is a variable used to denote the position of terms in a summation. It typically starts at a specified lower limit and increments by one until it reaches an upper limit. In the expression a + ar + ar^2 + ... + ar^12, the index can be represented by 'k', allowing us to express the series as Σ from k=0 to 12 of ar^k, where k indicates the exponent of r in each term.