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प्रश्न
Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them
`(3"n" - 2)/(3^("n" - 1))`
उत्तर
The given nth term is an = `(3"n" - 2)/(3^("n" - 1))`
a1 = `(3 xx 1 - 2)/(3^(1 - 1)) = (3 - 2)/3^0 = 1/1` = 1
a2 = `(3 xx 2 - 2)/(3^(2 - 1)) = (6 - 2)/3^1 = 4/3`
a3 = `(3 xx 3 - 2)/(3^(3 - 1)) = (9 - 2)/3^2 = 7/9`
a4 = `(3 xx 4 - 2)/(3^(4 - 1)) = (12 - 2)/3^3 = 10/27`
a5 = `(3 xx 5 - 2)/(3^(5 -1)) = (15 - 2)/3^4 = 13/81`
a6 = `(3 xx 6 - 2)/(3^(6 -1)) = (18 - 2)/3^5 = 16/243`
∴ The given sequence is `1, 4/3, 7/9, 10/27, 13/81, 16/243`
`1/1 (1/3)^0, 4(1/3)^1, 7(1/3)^2, 10(1/3)^3, 13(1/3)^4, 16(1/3)^5`
1, 4, 7, 10, 3, 16, ........... is in A.P.
`(1/3)0, 1/3, (1/3)^2, 1/3)^3, (1/3)^4, (1/3)^5` ........... is in G.P.
∴ The given sequence is A.G.P.
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