Evaluate the following summation: ∑ i = 1 4 i 2 − 3 i + 8 \sum_{i=1}^4i^2-3i+8

So in this problem, we're asked to evaluate the following summation. And doing this should be relatively straightforward. We're just taking this entire function that we have in here, this equation, and we're adding it to itself with different numbers plugged in. And we're starting from one, and then we're gonna go all the way up to four. So let's just go ahead and do that. Now I is the variable that we're looking at in this equation. So to do this summation, well, first off, we're going to take these i's and replace them with one. So it's going to give us one squared minus three times one plus eight. And we need to increment this all the way up to four. This is a summation, so we're adding it to the next piece. So we're going to have, and then that's gonna be one. In this case, it's gonna be two squared, and it's gonna be minus three times two plus eight. And then we're going to increment up again. So we're going to add, and it's going to be three squared, and it's gonna be minus three times three plus eight. And then we're going to finally go up to our last increment here, which stops at four. So we're going to go to four squared. It's gonna be minus three times four plus eight. So we've gone ahead and we've written out this entire sum right here. And you can find each of these pieces individually. Now you can crunch each of these numbers within the brackets yourself if you'd like to, and I'll just show you what these come out to in this video. So this is all gonna come out to six. And if you crunch these numbers, that'll come out to six as well. You need to square this, multiply three and a two, add eight to that result. Then you can run these numbers and that's going to give you eight, and you can run those numbers which will give you 12. So we have six plus six plus eight plus 12, which all comes out to 32 as our final result. So 32 is gonna be the solution to the sum, and that is how we can solve this problem. Hope you found this video helpful. Let's move on and get some more practice.

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0. Functions7h 55m

Introduction to Functions
18m

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

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 5m

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
1h 13m

Work
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11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 31m

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
52m

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

Slope Fields
19m

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22m

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1h 25m

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Sequences
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11m

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3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

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:
$\sum_{i = 1}^{4} i^{2} - 3 i + 8$

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Verified step by step guidance

Identify the expression inside the summation: $ i^2 - 3i + 8 $. This is the function we will evaluate for each integer value of $ i $ from 1 to 4.

Substitute $ i = 1 $ into the expression: $ 1^2 - 3(1) + 8 $. Calculate the result of this expression.

Substitute $ i = 2 $ into the expression: $ 2^2 - 3(2) + 8 $. Calculate the result of this expression.

Substitute $ i = 3 $ into the expression: $ 3^2 - 3(3) + 8 $. Calculate the result of this expression.

Substitute $ i = 4 $ into the expression: $ 4^2 - 3(4) + 8 $. Calculate the result of this expression. Finally, sum all the results from the previous steps to find the total value of the summation.