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Question
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
`(("n" + 1)("n" + 2))/(("n" + 3)("n" + 4))`
Solution
Given nth term an = `(("n" + 1)("n" + 2))/(("n" + 3)("n" + 4))`
a1 = `((1 + 1)(1 + 2))/((1 + 3)(1 + 4))`
= `(2*3)/(4*5)`
= `6/20`
= `3/10`
a2 = `((2 + 1)(2 + 2))/((2 + 3)(2 + 4))`
= `(3*4)/(5*6)`
= `12/30`
= `2/5`
a3 = `((3 + 1)(3 + 2))/((3 + 3)(3 + 4))`
= `(4*5)/(6*7)`
= `20/42`
= `10/21`
a4 = `((4 + 1)(4 + 2))/((4 + 3)(4 + 4))`
= `(5*6)/(7*8)`
= `30/56`
= `15/28`
a5 = `((5 + 1)(5 + 2))/((5 + 3)(5 + 4))`
= `(6*7)/(8*9)`
= `42/72`
= `7/12`
a6 = `((6 + 1)(6 + 2))/((6 + 3)(6 + 4))`
= `(7*8)/(9*10)`
= `56/90`
= `28/45`
∴ The given sequece is `3/10, 2/5, 10/21, 15/28, 7/12, 28/45`.
This is neither A.P, G.P nor AGP
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