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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 33
Chapter 9, Problem 33

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. ∑i=1105⋅2i\(\sum\)_{i=1}^{10} 5 \(\cdot\) 2^i

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1
Identify the terms of the geometric sequence. Here, the general term is given by \(a_i = 5 \cdot 2^i\), where \(i\) goes from 1 to 10.
Recognize that this is a geometric sequence with the first term \(a_1 = 5 \cdot 2^1 = 10\) and common ratio \(r = 2\) because each term is multiplied by 2 to get the next term.
Recall the formula for the sum of the first \(n\) terms of a geometric sequence: \(S_n = a_1 \cdot \frac{r^n - 1}{r - 1}\).
Substitute the known values into the formula: \(n = 10\), \(a_1 = 10\), and \(r = 2\), so the sum is \(S_{10} = 10 \cdot \frac{2^{10} - 1}{2 - 1}\).
Simplify the denominator and prepare to calculate the numerator \(2^{10} - 1\) to find the sum \(S_{10}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 5, 10, 20, 40, the common ratio is 2. Understanding this helps identify the pattern in the given sum.
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Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula simplifies adding many terms without computing each individually.
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Index and Exponent in Summation Notation

Summation notation (Σ) represents the sum of terms indexed by i from a starting value to an ending value. In this problem, the exponent i in 2^i changes with each term, affecting the value of each term in the sum. Understanding how the index affects each term is crucial for applying the sum formula correctly.
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