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Math901
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@MathMath901 The logic is a-b = cd
where, c = a+b and d = a-b
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@QED_Insights Lets do work in proving those aₙ limits using our old friend definition:
∀ ε > 0, ∃ N ∈ ℕ such that n > N ⇒ |aₙ − l| < ε
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Prove those limits using this definition, would be like to ask a programmer to create a web page using assembler.

Q.E.D Insights@QED_Insights
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@MathMath901 Actually it's all made up but to correspond with what we agreed upon it is not possible to simplify sqrt(-1)*sqrt(-1) to sqrt((-1)*(-1))
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@MathMath901 i is not sqrt(-1) as there is no sqrt of a negative value.
i^2 = -1 which is a different concept.
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@MathMath901 The 3rd line is invalid. In general Sqrt[a] X Sqrt[b] is not Sqrt[a b]. For positive a,b this is true. But not for negative a,b.
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Alternative ( System 1 thinking ,intuitive, fast pattern recognition) , since the four digit 9s, 6s and 0s only give 4, so 9, 6,0 ---> 1, and rest other terms give 0.
So, clearly, pick a term that consists of 8 but not 6,
like pick first 8809 = 6
=> 8+8 +1+1 =6
=> 2(8) = 4
Or 8 = 2
Thus, 2581 = 2+5+8+1 = 0+0+2+0 = 2 ( ans).
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Ans = 2,
Logic :
All terms are 4 digit, so those with all digits same here give either 0 or 4
So, those with 1,2,3,5,7 give 0, so each of them correspond to 0. But under an assumption that none of the terms are negative and all are whole numbers. Else there will be alternate positive and same negative numbers
And, those only with 0,6,9 give 4.
Means, 0000=4
6666=4
9999=4
This is possible iff all are 1 so they sum up to give 4.
So, this gives the value of 0,1,2,3,5,7 and 9.
Now, the remaining numbers left are 8 and 6 so choose any digit that consist of either 6 or 8 but not both together and rest all other terms.
One such term is 7662 =2, where say, 6 = X.
So, clearly 7+X+X+2 = 2.
So, 7--> 0 and 2-->0
Thus, 2X+0 =2
So, X = 1 or 6 =1
Now, see other terms that only consists of either 8 or 8 and 6 together.
Clearly, one such term is
8096 = 5
=> Say, 8 = Y
=> Y +0+9+6 = 5
=> Y + 1+1+1 = 5
=> Y + 3 = 5
=> Y =2 or 8 = 2
Hence, 2581 = 2+5+8+1 = 0+0+2+0 = 2
[Answer = 2]
Math901@MathMath901
#math problem 11-06-2026 Pattern recognition. What number corresponds to 2581?
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@MathMath901 A function can be exponential iff there is a variable in the power.
Thus, b,c and f.
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