Question

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

Answer 1

Using an arithmetic sequence, it is found that the smallest possible sum for the series is of 20 736, given by option B.

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• 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:

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

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• 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: $a_{384} = a_1 + 383(2) = a_1 + 766$

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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 $a_1$.

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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.

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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.

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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