Let a n , n ∈ N be an A.P. with common difference d ( d  = 0 ) and all terms non-zero. If n approaches infinity, then the sum a 1 a 2 1 + a 2 a 3 1 + ⋯ + a n a n + 1 1 will approa

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Let a n ​ , n ∈ N be an A.P. with common difference d ( d  = 0 ) and all terms non-zero. If n approaches infinity, then the sum a 1 ​ a 2 ​ 1 ​ + a 2 ​ a 3 ​ 1 ​ + ⋯ + a n ​ a n + 1 ​ 1 ​ will approach

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∑ = d 1 ​ ∑ a k ​ a k + 1 ​ a k + 1 ​ − a k ​ ​ = d 1 ​ ∑ ( a k ​ 1 ​ − a k + 1 ​ 1 ​ ) = d 1 ​ ( a 1 ​ 1 ​ − a n + 1 ​ 1 ​ ) . As n → ∞ , a n + 1 ​ = a 1 ​ + n d → ∞ in magnitude, so a n + 1 ​ 1 ​ → 0 . The sum tends to a 1 ​ d 1 ​ .

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