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Chapter 7: Problem 11

Find the sum of all the four-digit positive integers.

Short Answer

Expert verified
The sum of all the four-digit positive integers is 49,495,500.

Step by step solution

01

Determine the smallest and largest four-digit numbers

The smallest four-digit number is 1000, and the largest four-digit number is 9999.
02

Determine the number of terms

In order to determine the number of terms, we need to count how many numbers are there between 1000 and 9999, inclusive. We can do this by subtracting the smallest number from the largest number and add 1 to the result (because both extreme values are included): \(n = 9999 - 1000 + 1 = 9000\)
03

Apply the arithmetic series formula

Now that we have the first term (\(a_1 = 1000\)), the last term (\(a_n = 9999\)) and the number of terms (\(n = 9000\)), we can apply the arithmetic series formula to find the sum of all the four-digit positive integers: \[S_n = \frac{n(a_1 + a_n)}{2}\] \[S_{9000} = \frac{9000(1000 + 9999)}{2}\]
04

Calculate the sum

Plug in the values into the formula and calculate the sum: \[S_{9000} = \frac{9000(1000 + 9999)}{2} = \frac{9000(10999)}{2} = 9000 \cdot 5499.5 = 49495500\] The sum of all the four-digit positive integers is 49,495,500.

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Most popular questions from this chapter

Show that the sum of a finite arithmetic se- \(-\) quence is 0 if and only if the last term equals the negative of the first term.

Restate the symbolic version of the formula for evaluating a geometric series using summation notation.

Show that $$ \sqrt{n^{2}+n}-n=\frac{1}{\sqrt{1+\frac{1}{n}}+1} $$ [Hint: Multiply by \(\sqrt{n^{2}+n}-n\) by \(\left(\sqrt{n^{2}+n}+n\right) /\left(\sqrt{n^{2}+n}+n\right) .\) Then factor \(n\) out of the numerator and denominator of the resulting expression.]

Find the sum of all the four-digit odd positive integers.

Consider the sequence whose \(n^{\text {th }}\) term \(a_{n}\) is given by the indicated formula. (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Write the sequence as a recursive sequence. \(a_{n}=1-6 n\)
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