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Chapter 9: Problem 46

Find the sum using the formulas for the sums of powers of integers. $$\sum_{n=1}^{30} n$$

Short Answer

Expert verified
The sum of the first 30 natural numbers is 465.

Step by step solution

01

Identify the Formula

Identify the formula for the sum of the first \(n\) natural numbers, which is \(\ rac{n(n + 1)}{2}\}.
02

Substitute the Values

Substitute the value of \(n = 30\) into the formula: \(\ rac{n(n + 1)}{2}\} equals \(\ rac{30(30+1)}{2}\}.
03

Perform the Calculation

Perform the calculation. \(\ rac{30*31}{2}\) equals \(465\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Sum Formula
When it comes to summing up a list of numbers, you can use different formulas designed specifically for this purpose. One of the most frequently used is the formula for summing natural numbers. This formula, derived from mathematical principles, helps us quickly calculate the sum of an integer sequence without needing to add each number individually. The formula is \[s = \frac{n(n + 1)}{2} \]where **s** is the sum of the numbers and **n** is the total count of the natural numbers. This formula simplifies your task, whether you are summing the first 5 or the first 5000 natural numbers, making it a handy tool in mathematics.
Exploring Natural Numbers
Natural numbers are the counting numbers that start from 1 and go up infinitely. These include numbers like 1, 2, 3, and so on. Natural numbers are used in elementary counting and ordering, which makes them intuitive for most people. They have no fractions or decimals. Natural numbers form the base of many mathematical operations, including the sum of integers, and are foundational in the study of integer sequences. Understanding natural numbers is crucial since they form the initial set of integers from which more complex number systems are developed.
Investigating Integer Sequences
Integer sequences are ordered lists of integers that may follow specific patterns or rules. The sequence can be arithmetic, where each number differs from the previous one by a constant, or geometric, where each number is multiplied by a constant to get the next one. Sum formulas, like the one for natural numbers, help in efficiently handling integer sequences. These sequences can help model real-world problems, forecast patterns, and solve complex mathematical quests. Becoming familiar with integer sequences empowers you to tackle more intricate problems involving numbers and their arrangements.

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Most popular questions from this chapter

The amounts \(f(t)\) (in billions of dollars) of child support collected in the United States from 2002 through 2009 can be approximated by the model $$f(t)=-0.009 t^{2}+1.05 t+18.0, \quad 2 \leq t \leq 9$$, where \(t\) represents the year, with \(t=2\) corresponding to 2002. (Source: U.S. Department of Health and Human Services). (a) You want to adjust the model so that \(t=2\) corresponds to 2007 rather than \(2002 .\) To do this, you shift the graph of \(f\) five units to the left to obtain \(g(t)=f(t+5) .\) Use binomial coefficients to write \(g(t)\) in standard form. (b) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (c) Use the graphs to estimate when the child support collections exceeded \(\$ 25\) billion.

Prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n\).The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).

Complete the table and describe the result.$$\begin{array}{|c|c|c|c|} \hline n & r & _{n} C_{r} & _{n} C_{n-r} \\\\\hline 9 & 5 & & \\\\\hline 7 & 1 & & \\\\\hline 12 & 4 & & \\ \hline 6 & 0 & & \\\\\hline 10 & 7 & & \\\\\hline\end{array}$$,What characteristic of Pascal's Triangle does this table illustrate?

Simplify the difference quotient, using the Binomial Theorem if necessary.\(\frac{f(x+h)-f(x)}{h}\). $$f(x)=\frac{1}{x}$$

You are dealt five cards from a standard deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.)
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