Evaluate the following summation: ∑ k = 1 4 ( k 2 ) 2 \sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2

in this problem we are given this summation right here, and we are asked to evaluate the following sum. Now to do this, we are going to take a look at these values that we have in the lower and upper index, so we can see that we go from k equals one to zero of this equation at k over two squared. So to do this, well, we're gonna start from one, and we're gonna increment all the way up to four and add each result. That's how a summation works. So we'll start with a k value of one, which is gonna give us one over two squared. Then we're going to add our next k up value, which would be two. So we go one, two, three, four. So we're gonna go two over two. That's gonna be squared. Then we're going to go and then this is what? Three over two squared. Then we're going to finish off at four, which is going to give us four over two squared. Now what I can do is square each one of these results. So we're going to have one squared over two squared, which comes out to one over four. And here, we're going to have two squared over two squared, which comes out to four over four. And you could reduce this to just be one, but I'm gonna keep this in the denominator here to make things a little bit easier for us. So then we're going to have plus, and then we'll have three squared three over two squared. Three squared over two squared, which comes up to nine over four. And then lastly, we're gonna have four squared over two squared, which comes up to 16 over four. Now what you can do at this point is add all of these numbers together. You can add one plus four plus nine plus 16, and you should get 30. So you're gonna get 30 over four, which all reduces down to 15 over two. So this right here would be our result and the solution to this problem. That is how you can solve this sum. Let's move on.

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0. Functions7h 54m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

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27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
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Finding Global Extrema
50m

The First Derivative Test
1h 6m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

9. Graphical Applications of Integrals2h 27m

Area Between Curves
1h 18m

Introduction to Volume & Disk Method
1h 8m

10. Physics Applications of Integrals 3h 16m

Kinematics
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Work
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Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
55m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

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Basics of Differential Equations
48m

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19m

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22m

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1h 25m

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11m

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3h 17m

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Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
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Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

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Multiple Choice

Evaluate the following summation:
$\sum_{k = 1}^{4} {(\frac{k}{_{} 2})}^{2}$

$\frac{15}{2}$

$15$

$\frac{17}{2}$

$\frac{17}{4}$

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Verified step by step guidance

First, understand the summation notation: $ \sum_{k=1}^4 \left( \frac{k}{2} \right)^2 $. This means you will evaluate the expression $ \left( \frac{k}{2} \right)^2 $ for each integer value of $ k $ from 1 to 4, and then sum the results.

For $ k = 1 $, calculate $ \left( \frac{1}{2} \right)^2 $. This involves squaring the fraction $ \frac{1}{2} $.

For $ k = 2 $, calculate $ \left( \frac{2}{2} \right)^2 $. Simplify the fraction first, then square the result.

For $ k = 3 $, calculate $ \left( \frac{3}{2} \right)^2 $. Square the fraction $ \frac{3}{2} $.

For $ k = 4 $, calculate $ \left( \frac{4}{2} \right)^2 $. Simplify the fraction first, then square the result. Finally, sum all the squared values obtained from each step to find the total summation.