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प्रश्न
Find five numbers in G. P. such that their product is 1024 and fifth term is square of the third term.
उत्तर
Let the five numbers in G. P. be
`"a"/"r"^2, "a"/"r", "a", "ar", "ar"^2`
According to the given conditions,
`"a"/"r"^2 xx "a"/"r" xx "a" xx "ar" xx "ar"^2` = 1024
`"a"/cancel("r"^2) xx "a"/cancel("r") xx "a" xx "a"cancel("r") xx "a"cancel("r"^2)` = 1024
∴ a5 = 45
∴ a = 4 ...(i)
Also, ar2 = a2
∴ `"r"^2 = a^2/a`
∴ r2 = a
∴ r2 = 4 ...[From (i)]
∴ r = ± 2
When a = 4, r = 2
`"a"/"r"^2 = 1, "a"/"r" = 2`, a = 4, ar = 8, ar2 = 16
When a = 4, r = - 2
`"a"/"r"^2 = 1, "a"/"r" = - 2`, a = 4, ar = - 8, ar2 = 16
∴ The five numbers in G.P. are
1, 2, 4, 8, 16 or 1, - 2, 4, - 8, 16.
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