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in Binomial theorem by (48.3k points)
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If the mth, nth and pth terms of an AP and GP be equal and be, respectively, x, y, z. Then prove that  xy -z.yz - x.zx - y = 1 or xyyzzx = xz.yx.zy

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Let a be the first term and d be the common difference of the AP. Then

x = a + (m − 1)d, y = a + (n − 1)d, z = a + (p − 1)d

Let A be the first term and R be the common ratio of the GP. Then

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