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77julia77 [94]
3 years ago
6

A series of 384 consecutive odd integers has a sum that is a perfect fourth power of a positive

Mathematics
1 answer:
777dan777 [17]3 years ago
8 0

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:

a_n = a_1 + (n-1)d

  • The sum of the first n terms is given by:

S_n = \frac{n(a_1+a_n)}{2}

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

  • The set of odd integers is an arithmetic sequence with common difference 2, thus d = 2.
  • 384 terms, thus n = 384
  • The last term is: a_{384} = a_1 + 383(2) = a_1 + 766

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

The sum of the 384 terms is:

S_{384} = \frac{384(a_1 + a_1 + 766)}{2} = 192(2a_1 + 766) = 384a_1 + 147072

Now, for each option, we have to test if it generates an odd a_1.

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

Option d: Sum of 1296, thus, S_{384} = 1296, solve for a_1

384a_1 + 147072  = 1296

a_1 = \frac{1296 - 147072}{384}

a_1 = -379.6

Not an integer, so not the answer.

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

Option c: Test for 10000.

384a_1 + 147072  = 10000

a_1 = \frac{10000 - 147072}{384}

a_1 = -356.9

Not an integer, so not the answer.

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

Option b: Test for 20736.

384a_1 + 147072  = 20736

a_1 = \frac{20736 - 147072}{384}

a_1 = -329

Integer an odd, thus, option b is the answer.

A similar problem is given at brainly.com/question/16720434

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