Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Using an arithmetic sequence, it is found that the smallest possible sum for the series is of 20 736, given by option B.
---------------
- In an arithmetic sequence, the difference of consecutive terms is always the same, called common difference.
- The nth term of an arithmetic sequence is given by:
[tex]a_n = a_1 + (n-1)d[/tex]
- The sum of the first n terms is given by:
[tex]S_n = \frac{n(a_1+a_n)}{2}[/tex]
---------------
- The set of odd integers is an arithmetic sequence with common difference 2, thus [tex]d = 2[/tex].
- 384 terms, thus [tex]n = 384[/tex]
- The last term is: [tex]a_{384} = a_1 + 383(2) = a_1 + 766[/tex]
---------------
The sum of the 384 terms is:
[tex]S_{384} = \frac{384(a_1 + a_1 + 766)}{2} = 192(2a_1 + 766) = 384a_1 + 147072[/tex]
Now, for each option, we have to test if it generates an odd [tex]a_1[/tex].
---------------
Option d: Sum of 1296, thus, [tex]S_{384} = 1296[/tex], solve for [tex]a_1[/tex]
[tex]384a_1 + 147072 = 1296[/tex]
[tex]a_1 = \frac{1296 - 147072}{384}[/tex]
[tex]a_1 = -379.6[/tex]
Not an integer, so not the answer.
---------------
Option c: Test for 10000.
[tex]384a_1 + 147072 = 10000[/tex]
[tex]a_1 = \frac{10000 - 147072}{384}[/tex]
[tex]a_1 = -356.9[/tex]
Not an integer, so not the answer.
---------------
Option b: Test for 20736.
[tex]384a_1 + 147072 = 20736[/tex]
[tex]a_1 = \frac{20736 - 147072}{384}[/tex]
[tex]a_1 = -329[/tex]
Integer an odd, thus, option b is the answer.
A similar problem is given at https://brainly.com/question/16720434
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
