x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPTa)x = wb)x wc)x/y is an integerd)w/z is an integerCorrect answer is option 'C'. Can you explain this answer?

Question

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPTa)x = wb)x > wc)x/y is an integerd)w/z is an integerCorrect answer is option 'C'. Can you explain this answer?

Answer

= 2wc)w = 2xd)x = w + 1

To solve this problem, we need to use the formulas for the sum of consecutive integers. The sum of y consecutive integers is given by:

x = n + (n+1) + (n+2) + ... + (n+y-1) = yn + y(y-1)/2

where n is the first integer in the sequence.

Similarly, the sum of z consecutive integers is given by:

w = m + (m+1) + (m+2) + ... + (m+z-1) = zm + z(z-1)/2

where m is the first integer in the sequence.

Using the fact that y = 2z, we can rewrite the formula for x as:

x = 2zn + 2z(z-1)/2 = z(2n+z-1)

Now we can consider each answer choice:

a) x = w

If x = w, then we have:

z(2n+z-1) = zm + z(z-1)/2

Simplifying this equation, we get:

2n + z - 1 = m + z/2 - 1/2

2n - m = z/2

But this means that z is even, which contradicts the assumption that y and z are both positive integers. Therefore, this option cannot be true.

b) x = 2w

If x = 2w, then we have:

z(2n+z-1) = 2zm + 2z(z-1)/2

Simplifying this equation, we get:

2n + z - 1 = 2m + z - 1

2n = 2m

This implies that n = m, which means that the sequences of consecutive integers are the same. Therefore, this option could be true.

c) w = 2x

If w = 2x, then we have:

zm + z(z-1)/2 = 2z(n + (n+1) + ... + (n+2z-1))

Simplifying this equation, we get:

m + z/2 - 1/2 = 2n + 2z - 2

m - 2n = z/2 + 3/2

But this means that z is odd, which again contradicts the assumption that y and z are both positive integers. Therefore, this option cannot be true.

d) x = w + 1

If x = w + 1, then we have:

z(2n+z-1) = zm + z(z-1)/2 + 1

Simplifying this equation, we get:

2n + z - 1 = m + z/2

2n - m = z/2 - 1/2

Again, this implies that z is even, which is not possible. Therefore, this option cannot be true.

In conclusion, the only option that could be true is b) x = 2w.

Answer

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