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प्रश्न
For a G.P., if the sum of the first 3 terms is 125 and the sum of the next 3 terms is 27, find the value of r.
उत्तर
S3 = 125, S6 = 125 + 27 = 152
Sn = `"a"((1 - "r"^"n")/(1 - "r"))`
∴ S3 = `"a"((1 - "r"^3)/(1 - "r"))`
∴ 125 = `"a"((1 - "r"^3)/(1 - "r"))` ...(i)
Also, S6 = `"a"((1 - "r"^6)/(1 - "r"))`
∴ 152 = `"a"((1 - "r"^6)/(1 - "r"))` ...(ii)
Dividing (ii) by (i), we get
`152/125 = (1 - "r"^6)/(1 - "r"^3)`
∴ `152/125 = ((1 + "r"^3)(1 - "r"^3))/((1 - "r"^3)`
∴ 1 + r3 = `152/125`
∴ r3 = `152/125 - 1`
∴ r3 = `27/125`
∴ r3 = `(3/5)^3`
∴ r = `3/5`
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