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What is the sum of the coefficients of the expansion of (5a/3-2b/3)^10?

Jul 14, 2023

#1
-3

The sum of the coefficients of the expansion of (5a/3-2b/3)^10 is 1024.

The binomial theorem tells us that the sum of the coefficients of the expansion of (x+y)^n is given by 2^n. In this case, x = 5a/3 and y = -2b/3, so the sum of the coefficients is 2^10 = 1024.

Here is a proof of the binomial theorem:

(x+y)^n = nC0x^n + nC1x^(n-1)y + nC2x^(n-2)y^2 + ... + nCy^(n-1)x + y^n

The sum of the coefficients is given by

nC0 + nC1 + nC2 + ... + nCy^(n-1) + y^n

We can use Pascal's identity to simplify this expression:

nC0 + nC1 + nC2 + ... + nCy^(n-1) + y^n = (nC0 + y^n) + (nC1 + y^(n-1)) + ... + (nCn-1)

Pascal's identity tells us that nC0 + y^n = 2^n, so the sum of the coefficients is

2^n + (nC1 + y^(n-1)) + ... + (nCn-1)

This expression is always equal to 2^n. Therefore, the sum of the coefficients of the expansion of (x+y)^n is always 2^n.

In the case of (5a/3-2b/3)^10, x = 5a/3 and y = -2b/3, so the sum of the coefficients is 2^10 = 1024.

Jul 14, 2023
#2
+4

This is an example of a chat-gpt answer. Does it sound like chatGPT? Yes. Does ZeroGPT think it is? Yes. Is it correct? No. Is moderation going to do anything? No. "B-but we don't have any pwoof!!" I don't care. It's incorrect anyhow, and should be deleted. Of course, we are keen on protecting the egos of anonymous users more than the quality of the site.

Guest Jul 14, 2023
#5
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(5a/3  -  2b/3)^10?

(5/3  -  2/3) =1 - sum of the coefficients

Jul 15, 2023
#6
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