A series of 288 consecutive odd integers has a non-zero sum that is a perfect fourth power. Find the smallest possible sum for this series.
a series of 288 consecutive odd integers has a non-zero sum that is a perfect fourth power, find the smallest possible sum for this series? (I hope I’m right idk how to explain this tho)
a = smallest odd integer
S = a + (a+2) + (a+4) + ... + (a+2*287) = 288(a+(a+574))/2 = 288(a+287)
288 = 2^5 * 3^2
Therefore, smallest value of (a+287) that will make S = 288(a+287) a perfect fourth power is:
a + 287 = 2^3 * 3^2 = 72
a = −215
Smallest sum = 288*72 = 20736 = 12^4
Smallest odd integer in series = −215
−215 + −213 + −211 + ... + −1 + 1 + 3 + ... + 213 + 215 + ... + 359
= 20736 = 12^4
whos right
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thank you<3
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bhbiub
To find the smallest possible sum for the series of consecutive odd integers, we need to determine the length of the series and the starting point. Let's break down the problem step by step:
Step 1: Determine the length of the series
Since the series consists of consecutive odd integers, we know that the difference between any two consecutive terms is 2. To find the length, we can divide the total difference in the series by 2.
The total difference in the series can be found by subtracting the first term from the last term: (last term) - (first term) = total difference.
Let's represent the first term as a and the last term as b. Since there are 288 terms in the series, we have:
total difference = b - a = 288 * 2 = 576
Step 2: Find a suitable fourth power
We need to find a suitable fourth power whose non-zero sum can be obtained by adding consecutive odd integers. We can start by testing small fourth powers:
1^4 = 1, 2^4 = 16, 3^4 = 81, ... and so on.
Let's check if any of these fourth powers can be represented as a non-zero sum of consecutive odd integers.
Step 3: Test the potential fourth power value
Let's consider the potential fourth power as c.
We can express the sum of the series as:
Sum = a + (a + 2) + (a + 4) + ... + b
= ((b - a) / 2 + 1) / 2 * (a + b)
= (288/2) * (a + b)
= 144(a + b)
We need to find the values of a and b such that the sum is equal to c^4.
So, we have the equation:
c^4 = 144(a + b)
Step 4: Solve the equation
Substituting the value of the total difference (576) into the equation, we get:
c^4 = 144(576)
c^4 = 82944
Taking the fourth root of both sides, we get:
c = ∛(82944)
By solving this equation, we find that c ≈ 42.46397.
Step 5: Find the values of a and b
To find the smallest possible sum for this series, we need to find the smallest values of a and b that satisfy the equation.
Divide the total difference (576) by 2:
576 / 2 = 288
Now, subtract half of the total difference from the potential fourth power:
b = c - 144
b ≈ 42.46397 - 144
b ≈ -101.53603
To find a, add half of the total difference to b:
a = b + 288
a ≈ -101.53603 + 288
a ≈ 186.46397
Since we are dealing with consecutive odd integers, we need to adjust the values of a and b to the nearest odd integers. Let's round down a to 185 and round up b to -101.
Therefore, the smallest possible sum for the series is:
Sum = a + (a+2) + (a+4) + ... + b
Sum = 185 - 101
Sum = 84
If the sum starts at the (k+1)st odd number, its sum is
(k+288)^2 - k^2 = 576k + 82944
When k=432, the sum is 331,776 = 24^4
How you'd find that algebraically I have no clue.