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Ch. 5 - Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 5 - IntegralsProblem 5.PE.3a
Chapter 5, Problem 5.PE.3a

           10           10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
           k = 1          k = 1


  10
a. Σ aₖ/4
  k = 1

Verified step by step guidance
1
Identify the given information: \( \sum_{k=1}^{10} a_k = -2 \) and \( \sum_{k=1}^{10} b_k = 25 \).
Understand that the problem asks for \( \sum_{k=1}^{10} \frac{a_k}{4} \), which means each term \( a_k \) is divided by 4 before summing.
Recall the property of summations that allows factoring out constants: \( \sum_{k=1}^{n} c \cdot a_k = c \cdot \sum_{k=1}^{n} a_k \), where \( c \) is a constant.
Apply this property to the given sum: \( \sum_{k=1}^{10} \frac{a_k}{4} = \frac{1}{4} \sum_{k=1}^{10} a_k \).
Substitute the known sum \( \sum_{k=1}^{10} a_k = -2 \) into the expression to write \( \sum_{k=1}^{10} \frac{a_k}{4} = \frac{1}{4} \times (-2) \).

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Key Concepts

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

Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms indexed by an integer variable. It concisely expresses adding terms like a₁ + a₂ + ... + aₙ. Understanding the limits of summation and the general term is essential for manipulating and evaluating sums.
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Properties of Summations

Summations have linearity properties, meaning the sum of a constant times a sequence equals the constant times the sum of the sequence. For example, Σ (c * aₖ) = c * Σ aₖ. This property allows factoring constants out of sums, simplifying calculations involving scaled sequences.
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Given Series Sums and Their Use

Knowing the total sums of sequences (like Σ aₖ = -2) allows direct substitution when evaluating related sums. This is useful when the problem asks for sums of scaled or transformed sequences, enabling quick computation without summing individual terms.
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Related Practice
Textbook Question

Definite Integrals


In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.


     n

 lim  ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]

 ∥P∥→0  k = 1

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Textbook Question

If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.


b. ∫₁² g(x) dx

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Textbook Question

In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.

ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

Textbook Question

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

y² = 4x, y = 4x - 2

Textbook Question

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Evaluate the integrals in Exercises 47–68.


∫₋₁¹ (3x² - 4x + 7)dx

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Textbook Question

Evaluate the integrals in Exercises 37–46.


∫(2θ + 1 + 2 cos (2θ + 1))dθ