Question

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

Answer 1

The smallest sum is 20 736!

Explanation:

Hope this helps!!! :))

Answer 2

Explanation:

the 1st number of this series is x

because this series has 384 consecutive odd integers => this is a arithmetic sequence with d = 2

=> the sum of this series =
`384x+384(384-1) =384x+147072`

but this series has a sum that is the perfect fourth power of a positive integer

=> 384x + 147072 = k^4 ( k is a positive integer)

to have the smallest sum => x must be the smallest odd integer.

+) with the sum = 104 976 => x isnt a integer => unreasonable

+) with the sum = 20 736 => x = -329

+) with the sum = 10000 =>x isnt a integer => unreasonable

+) with the sum = 1296 => x isnt a integer => unreasonable

+) with the sum = 38 416 => x isnt a integer => unreasonable

=> the smallest sum is 20 736