Part 2: Applying the Formula for the Sum of the First n integers as shown in 5.2. Make up an original problem similar to example 5.2.2 Evaluate a. Evaluate 2+ 4 + 6 + ... + 200. b. Evaluate 1+ 2 + 3+4+ ... + 300.c. 5+7+9+11+...+159. b. Evaluate 1+ 2 + 3+4+ ... + 300. c. ### Theorem 5.2.1: Sum of the First n IntegersFor every integer $ n \geq 1 $,\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]---### DefinitionIf a sum with a variable number of terms is shown to equal an expression that does not contain either an ellipsis or a summation symbol, we say that the sum is written in **closed form**.---This content explains a fundamental theorem and associated definition in mathematics, particularly on the topic of summing the first $ n $ integers. The theorem provides a formula to find the sum of the first $ n $ natural numbers efficiently, which is represented in a closed form as $ \frac{n(n+1)}{2} $.

According to the given information it is required to find the sum of first n integers. For part (a)…

Part 2: Applying the Formula for the Sum of the First n integers as shown in 5.2. Make up an original problem similar to example 5.2.2 Evaluate a. Evaluate 2+ 4 + 6 + ... + 200. b. Evaluate 1+ 2 + 3+4+ ... + 300. c. 5+7+9+11+...+159. b. Evaluate 1+ 2 + 3+4+ ... + 300. c.

Part 2: Applying the Formula for the Sum of the First n integers as shown in 5.2. Make up an original problem similar to example 5.2.2 Evaluate a. Evaluate 2+ 4 + 6 + ... + 200. b. Evaluate 1+ 2 + 3+4+ ... + 300. c. 5+7+9+11+...+159. b. Evaluate 1+ 2 + 3+4+ ... + 300. c.

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.7: Problem Solving: Consecutive Integers

Problem 14OE

See similar textbooks

Related questions

Q: Use the formula n(n +1)(2n + 1) 6 r=1 to compute the sum 20 S= Σ. r=1

Q: We can use sigma notation to rewrite the sum 4 + 7+ 10 + 13 + + 31 as: ... 4 +7+ 10 + 13 + ..+ 31 =…

Q: first

Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: #3

Q: Compute the following sum and show your work. 1 2 + -1 3 1 2 -1 2 -1 Can you compute the following…

A: A set of numbers or functions arranged in the form of a rectangular array bounded by the square…

Q: n 2 (4k – 2) = k=1

A: Recall: 1+2+3+...+n=n(n+1)2

Q: Use the given table to write an explicit and a recursive rule for the sequence. 2. 3. 1 3 4 1 2 3 4…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

Q: Use the formula n(n+ 1)(2n + 1) 6 to compute the sum 20 S= 2r*. r=1

A: we are given the series ∑r=1nr2=n(n+1)(2n+1)6

Q: Write out the first 10 terms of each of the following sequences (starting with ao). Then give either…

A: (a) an = 3n + 4For n = 0,

Q: Consider the sequence {ax}, defined by а, 3D 1, ад — 1, and ak 3 а-1+ ak-2 for k > 3. Compute as.

A: We have given two terms of sequence {a_k}, a_1=1 and a_2=1 and then sequence {a_k} is defined by…

Q: Write the first four elements of the sequence. n+1 3n -1 O -1, 1, O 0, 1 1 3 2'4 O1.1 3 2 3'2' 4'3 O…

A: The given sequence is n+13n-1. The objective is to obtain the first four elements of the given…

Q: A sequence is defined by the explicit formula z, = 32(3)(n-1), What is the recursive formula for the…

A: given that zn=323n-1 find the recursive formula for the given sequence

Q: Write an explicit formula for the sequence: 1 1 11'12' 1 1 9' 10 1 13

Q: Suppose Suppose a, = 3n? – 3n – 3. Find a closed formula for the sequence of differences by…

A: Given, an = 3n2 -3n +3 We have to find an-an-1

Q: Find the sum of the sequence. I (k° +4) k=1

A: Given sequence is

Q: Write an explicit formula for the sequence: 1 1 12' 13 -1 1 8'9' 1 1 | .. 10' 11

Q: Find an explicit formula to calculate the sum (assuming n is a multiple of 2) P = 1 · 1 + 2 · 2 +…

Q: Write an explicit formula for the sequence 14 19 8 29 9, 3'7'5'31 5 n+ 4 2"-1 9 n b. an 2" -1 5n +4…

Q: Write an explicit formula for the sequence 19 22 25 16 13, 3' 7' 15' 31 3 n + 10 а. an 2" +1 3 n+ 10…

Q: 3n+2 4n–2° n=1

Q: 5(п+1) d. an 2n-1 е. ап 5n + 12 II

A: Given: an=5(n+1)2n-1 & an=5n+12 To determine: The first 10 terms of the given sequences.

Q: 33. Study the sequence (un) defined by 3 (a) uo = a E R, Vn E N : un+1 = un.

A: Since you have asked multiple questions, we will solve first question for you. If you want other…

Q: Compute the sum: n² – 3n – 1

Q: The sequence (sn) is defined recursively by s0=3 and sk=sk-1+2k. Show that sn=3+n(n+1) using…

A: We are going use method of mathematical induction to prove the result

Q: 9. Find the explicit formula for a sequence satisfying а, %3D10а,, + 25а, _,n22 n-1 а, %3D 3, а,…

Q: Find the third term of the sequence {2n^2/3^n-1}from n=1 to ∞

Q: Consider the sequence (am)neN defined by 1 A2m -1 for all m E N. A2m-1 m m ||

A: Given that (an)n∈N be the sequence defined asa2m-1=1ma2m=-1m

Q: Find the recurrence relation of the sequence defined by T(1) = 2 T(n) = 2n * T(n-1), n > 1

Q: what is the summation where k = 1 of e^-2k

Q: Give an explicit formula for an, the apparent nth term in the sequence 7, 17, 7, 17, 7, 17...

A: .

Q: Use the approach in Gauss’s Problem to find the following sums of arithmetic sequences. a. 1 + 2…

Q: Suppose young Gauss had been asked to compute the sum 6 + 12 + 18 + 24 + ... + 1,494 + 1,500 1. How…

Q: The generating function for the sequence 1,0.1,0.1,0,1 is 3)1 +x + x + + x+ x5 b)1 +x2 - 4. )1+ x+ x…

A: generating function is a way of encoding an infinite sequence of numbers (an) by treating them as…

Q: The five first terms of the sequence {4n–1}=1 are: O 3,7,11,15,19,... O 3+7+11+15+ 19... O…

Q: Compute the summation. 5 E (k + 3) = k = 1

Q: Use the approach in Gauss's Problem to find the following sums of arithmetic sequences.…

Q: If a sequence is defined by ( U1 = (Иn+2 3D 5иn+1 — 6ип for n> 2' 11, u, = 37 then the explicit…

A: Given sequence u1=11, u2=37 and for n>2, un+2=5un+1-6unThe sequence for n>2, can also be…

Q: There is a simple formula for the sum of the first n cubes. n' (n + 1)2 n i=1 Using this formula,…

A: Sum of cubes

Q: A sequence {an} is defined as follows: ao = 2, a1 = 1, and for n > 2, a, = 3 an-1 – n. an-2 + 1 %3D…

Q: A sequence is said to be an increasing sequence if Vn 21 ......... n+1 a O 25,00 n+1 zn.c O n+1 S =S…

Q: Compute for the sum. Answer directly. Lore ΣΣ (-)) i=1j=1

Q: {- Which of the following formula is the general term of the sequence 1 2 4 9 4. 16 25 .. 3 72 O a.…

A: We have to choose the correct answer from the given condition

Q: Use the approach in Gauss's Problem to find the following sums of arithmetic sequences. A.…

A: Consider the series 1+2+3+4+…+1001. For the series 1+2+3+4+…+1001, it is clear that there are 1001…

Q: Use the approach in Gauss’s Problem to find the following sums of arithmetic sequences. a.…

Q: 1.)Prove that every n ≥ 14 ∈ n can be written as the sum of sequences of 3's and 8's

Q: Question 2. Suppose that a; is a sequence of integers such that a; → L as i → o. Does L have to be…

A: We will use basic knowledge of real analysis and solve it using the epsilon definition. The claim is…

Q: Use the approach in Gauss's Problem to find the following sums of arithmetic sequences. a. 1+2+3+4+.…

Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Part 2: Applying the Formula for the Sum of the First n integers as shown in 5.2. Make up an original problem similar to example 5.2.2

Evaluate

a. Evaluate 2+ 4 + 6 + ... + 200.

b. Evaluate 1+ 2 + 3+4+ ... + 300.

c. 5+7+9+11+...+159.

b. Evaluate 1+ 2 + 3+4+ ... + 300.

### Theorem 5.2.1: Sum of the First n Integers

For every integer $ n \geq 1 $,

\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]

---

### Definition

If a sum with a variable number of terms is shown to equal an expression that does not contain either an ellipsis or a summation symbol, we say that the sum is written in **closed form**.

---

This content explains a fundamental theorem and associated definition in mathematics, particularly on the topic of summing the first $ n $ integers. The theorem provides a formula to find the sum of the first $ n $ natural numbers efficiently, which is represented in a closed form as $ \frac{n(n+1)}{2} $.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.