# Fuzzy Operators

$S_n = 1 + 2 - 3 \times 4\div 5 + 6 - 7 \times 8 \div 9 +10- \cdots n$

Consider the above sum where the first $$n$$ positive integers are listed in ascending order, and the mathematical operations $$(+,-,\times,\div)$$ are repeated in that order.

Find the value of $$S_{31417}$$. Give your answer to 3 decimal places.

Details and Assumptions:

• BODMAS (Order of Operations) is obeyed.

• A couple of explicit examples: \begin{align} S_4 &= 1 + 2 - 3\times4 = (1+2) - (3\times4) = 3 - 12=-9 \\ S_5 &= 1+2-3\times4\div5 = (1+2) - (3\times4\div5) = 3 - 12\div5 = 0.6. \end{align}

×