Problem 46 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 5+5^{2}+5^{3}+\cdots+5^{12} $$

Step-by-Step Solution

Answer

The sum can be expressed as \( \sum_{i=1}^{12} 5^{i} \).

Step 1: Identify the geometric series

The given expression is a geometric series. This is recognised because every term in the sequence is the product of the previous term and the constant 5. Thus the ratio from one term to the next term is always 5.

Step 2: Identify the variable

The question asks us to use \( i \) as the index of summation. We can denote the exponent of 5 as \( i \). So when \( i = 1 \), we have \( 5^{1} = 5 \), when \( i = 2 \), we have \( 5^{2} = 25 \), and so on.

Step 3: Write in summation notation

The problem asks to express this sum using summation notation with 1 as the lower limit of summation and \( i \) for the index of summation. We write this as: \( \sum_{i=1}^{12} 5^{i} \)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. For example, in the series
\[ 5 + 5^2 + 5^3 + \.\.\. + 5^{12} \],
the ratio is 5, because each term is 5 times the term before it. This type of series can be expressed in a more compact form using summation notation, which is incredibly useful for both simplifying expressions and for computation purposes. When dealing with geometric series, it is crucial to identify the first term and the common ratio, as they play a key role in determining the sum of the series or expressing the series in summation notation.

Index of Summation

The index of summation is typically represented by the letter 'i' and it serves as a placeholder that runs from the lower limit to the upper limit of the summation. In the given series,
\[ 5 + 5^2 + 5^3 + \.\.\. + 5^{12} \],
we are asked to use
\( i \)
as the index of summation with 1 as the lower limit. This means \( i \) starts at 1 and increases by 1 until it reaches 12. Each value of \( i \) corresponds to a term in the series, with the exponent on 5 matching the value of \( i \) at each step. By defining the index of summation, we can rewrite the series neatly and succinctly, making it easier to handle algebraically.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. In our example, we see numbers like \( 5^2 \) and \( 5^{12} \), where 2 and 12 are the exponents. The expression \( 5^i \) indicates that the number 5 is to be multiplied by itself as many times as indicated by \( i \), the index of summation.

Exponents are fundamental in working with geometric series, as they determine the terms of the series. Understanding how they work enables students to grasp why each term of the series is the result of raising the base (in this case, 5) to the power of the series' index of summation. This concept is used extensively in mathematics, from simple computations to complex functions.