3*2^x+1=y^2, where x and y are non-negative integers. Find the sum of

The question states \(3*2^x+1=y^2\) which means y^2, i.e, y has to be odd...
0 is non negative, so when x is 0...\(3*2^x+1=3*2^0+1=3*1+1=3+1=4=y^2...y=2\)..
Here y cannot be -2 as it is non negative

chetan2u Thank-you sir for correcting me. I didn't infer the question stem and its possibilities fully.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since we have 2 variable (\(x\) and \(y\)) and 1 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
If \(x = 0\), then we have \(3 \cdot 2^x+1=3 \cdot 2^0+1=4=y^2\) or \(y=2\) and so \(x + y = 2\).
If \(x = 4\), then we have \(3 \cdot 2^x+1=3 \cdot 2^4+1=49=y^2\) or \(y=7\) and so \(x + y = 11\).
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since \(y\) is an even integer, \(3 \cdot 2^x\) must be an odd integer and the only possible non-negative integer of \(x\) is \(0\).
If \(x = 0\), then we have \(3 \cdot 2^x+1=3 \cdot 2^0+1=4=y^2\) or \(y=2\) and so \(x + y = 2\).
Since condition 2) yields a unique solution, it is sufficient.

Therefore, B is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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