348927088idn
Answered

Connect with a global community of experts on IDNLearn.com. Get accurate and comprehensive answers to your questions from our community of knowledgeable professionals.

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 :

Caylusidn

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 value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.