Problem 19 \(\sum_{i=1}^{n}\left(10^{i+1}-1... [FREE SOLUTION]

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Chapter 7: Problem 19

\(\sum_{i=1}^{n}\left(10^{i+1}-10^{i}\right)\)

Short Answer

Expert verified

The series sums to \(10^{n+1} - 10\)

Step by step solution

Understanding the Series

Identify the given series. The expression to be summed is \(鈭慱{i=1}^{n}(10^{i+1} - 10^i)\)

Simplifying the Summand

Simplify the term inside the summation. \(10^{i+1} - 10^i = 10 \cdot 10^i - 10^i = 9 \cdot 10^i\)

Rewriting the Series

Rewrite the series using the simplified term: \(鈭慱{i=1}^{n} 9 \cdot 10^i\)

Factoring Out Constants

Factor out the constant 9 from the summation: \(9 \cdot 鈭慱{i=1}^{n} 10^i\)

Sum of a Geometric Series

Recognize that \(鈭慱{i=1}^{n} 10^i \) is a geometric series with the first term \(a = 10\) and the common ratio \(r = 10\). The sum of the first \(n \) terms of a geometric series is given by the formula: \( S_n = a \cdot \frac{r^n - 1}{r - 1}\)

Applying the Geometric Series Formula

Substitute \(a = 10\) and \(r = 10\) into the formula: \(S_n = 10 \cdot \frac{10^n - 1}{10 - 1} = \frac{10^{n+1} - 10}{9}\)

Combining Results

Substitute the result into the factored series: \(9 \cdot 鈭慱{i=1}^{n} 10^i = 9 \cdot \frac{10^{n+1} - 10}{9} = 10^{n+1} - 10\)

Final Answer

The final simplified form of the given summation is: \(鈭慱{i=1}^{n} (10^{i+1} - 10^i) = 10^{n+1} - 10\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Summation Notation

The notation \(鈭� \) is used to denote the sum of a sequence of terms. In this case, it sums the expression \(10^{i+1} - 10^i\) from \(i=1\) to \(n\). Summation notation is a concise way to represent the sum of a series of numbers. It specifies:

A variable that changes value within a specified range (here, \(i\) from \(1\) to \(n\)).
An expression that typically depends on that variable (here, \(10^{i+1} - 10^i\)).

Summation notation is practical because it clearly indicates the scope and the components of a series sum. Once you understand how to interpret this notation, it becomes an efficient tool for representing complex sums.

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 constant called the common ratio. For a geometric series with terms \(a, ar, ar^2, ar^3,...\), the sum of the first \(n\) terms, \(S_n\), is given by the formula: \[ S_n = a \cdot \frac{r^n - 1}{r - 1} \] Here:

\(a\) is the first term
\(r\) is the common ratio

In our example, the simplified series, \(鈭慱{i=1}^{n} 10^i\), is a geometric series with the first term \(a = 10\) and the common ratio \(r = 10\). Understanding geometric series allows us to efficiently calculate the sum of such series using the provided formula.

Series Simplification

Simplifying a series involves rewriting it to make it easier to compute. For the given series \(鈭慱{i=1}^{n} (10^{i+1} - 10^i)\), we simplify the expression inside the sum:

First, note that \(10^{i+1}\) can be written as \(10 \cdot 10^i\).
So, the term \(10^{i+1} - 10^i\) becomes \(10 \cdot 10^i - 10^i = 9 \cdot 10^i\).

Thus, the original series \(鈭慱{i=1}^{n} (10^{i+1} - 10^i)\) simplifies to \(鈭慱{i=1}^{n} 9 \cdot 10^i\), which is much easier to handle. By simplifying the series, we reduce complexity and make it possible to apply known formulas and techniques.

Factoring Constants

Factoring out constants from a series is a common and useful technique to simplify calculations. In the series \(鈭慱{i=1}^{n} 9 \cdot 10^i\), the constant \((9)\) can be factored out:

This leaves us with \(9 \cdot 鈭慱{i=1}^{n} 10^i\).

Factoring out constants streamlines the summation process by isolating the variable-dependent part of the series. This allows us to use summation formulas more effectively and can simplify further algebraic manipulations. By recognizing and factoring out constants, we make our mathematical work less cumbersome and more straightforward.

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