348927088
Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Discover a wealth of knowledge from experts across different disciplines on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Help please!!
A series of 384 consecutive odd integers has a sum that is the perfect fourth power of a positive integer. Find the smallest possible sum for this series.
a) 104 976
b) 20 736
c) 10 000
d) 1296
e) 38 416

Sagot :

caylus

Answer:

Step-by-step explanation:

Let say 2*a+1 the first odd integer.

[tex]n_1=2a+1\\n_2=2a+1+2=2a+3=2a+2*1+1\\n_3=2a+5=2a+2*2+1\\n_4=2a+2*3+1\\....\\n_{384}=2a+2*383+1=2a+767\\\\Sum=(2a+1)+(2a+3)+(2a+5)+...+(2a+767)\\=384*2a+(1+3+...+767)\\=384*2a+\dfrac{1+767}{2} 384\\=384*2a+384* 384\\=384*(2a+384)\\\\The\ sum\ must\ be \ divisible \ by\ 384.\\In\ the\ given\ numbers,\ only\ 20736\ is\ divisible\ by\ 384.\\\\20736=54*384\\\\2a+384=54\\2a=54-384\\2a=-330\\a=-165\\[/tex]

[tex]The\ first\ number\ is\ 2*a+1=2*(-165)+1=-329\\The\ last\ number\ is\ 2a+767=2*(-165)+767= 437\\-329-327-325-....+1+3+5+...+437=20736\\[/tex]

Proof in the file jointed.

View image caylus

Using an arithmetic sequence, the smallest possible sum is of 20 736, given by option b.

----------------------------

The nth term of an arithmetic sequence is given by:

[tex]a_n = a_1 + (n-1)d[/tex]

In which

  • The first term is [tex]a_1[/tex]
  • The common ratio is [tex]d[/tex].

The sum of the first n terms of a sequence is given by:

[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]

----------------------------

  • 384 means that [tex]n = 384[/tex], thus:

[tex]S_n = \frac{384(a_1 + a_n)}{2}[/tex]

[tex]S_n = 192(a_1 + a_n)[/tex]

  • Consecutive odd integers means that the common ratio is 2, thus [tex]d = 2[/tex], and the nth term is:

[tex]a_n = a_1 + (n-1)d = a_1 + 383(2) = a_1 + 766[/tex]

Thus, the sum is:

[tex]S_n = 192(a_1 + a_n)[/tex]

[tex]S_n = 192(a_1 + a_1 + 766)[/tex]

[tex]S_n = 384a_1 + 147072[/tex]

----------------------------

  • Now, for each sum, it is tested if [tex]a_1[/tex] is an odd integer.

For a sum of 104 976.

[tex]S_n = 384a_1 + 147072[/tex]

[tex]104976 = 384a_1 + 147072[/tex]

[tex]384a_1 = 104976 - 147072[/tex]

[tex]a_1 = \frac{104976 - 147072}{384}[/tex]

[tex]a_1 = -109.6[/tex]

Not an integer, so this option is not correct.

----------------------------

For a sum of 20 736.

[tex]S_n = 384a_1 + 147072[/tex]

[tex]20736 = 384a_1 + 147072[/tex]

[tex]384a_1 = 20736 - 147072[/tex]

[tex]a_1 = \frac{20736 - 147072}{384}[/tex]

[tex]a_1 = -329[/tex]

Odd integer, thus, the correct option is b.

A similar problem is given at https://brainly.com/question/24555380

We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.