Find a formula for the nth term of the sequence.
-0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated)
A) $a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }$
B) $a _ { n } = 1 + ( - 1 ) ^ {{ n } ( n - 1 ) / 2}$
C) $a _ { n } = 1 + ( - 1 ) ^ {{ n } ( n + 1 ) ( n + 2 ) / 6}$
D) $a _ { n } = 1 + ( - 1 ) ^ { ( n + 1 ) ( n + 2 ) / 2 }$

Question

Find a formula for the nth term of the sequence.
-0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated) 
A) $a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }$
B) $a _ { n } = 1 + ( - 1 ) ^ {{ n } ( n - 1 ) / 2}$
C) $a _ { n } = 1 + ( - 1 ) ^ {{ n } ( n + 1 ) ( n + 2 ) / 6}$
D) $a _ { n } = 1 + ( - 1 ) ^ { ( n + 1 ) ( n + 2 ) / 2 }$

Quizplus · Accepted Answer

The sequence alternates between 0 and 2 every 4 terms, and there are 3 2's in every 4 terms. Using modular arithmetic, we can express this as: 
$a_n = \begin{cases} 0, &	ext{if } n\equiv 0,4 \pmod 4\ 2, &	ext{if } n\equiv 1,2,3 \pmod 4\end{cases}$ 
Simplifying this expression using the Chinese Remainder Theorem, we get: 
$a_n = 1 + (-1)^{(n(n+1)(n+2)/6)}$ 
Therefore, the answer is C.

Find a Formula for the Nth Term of the Sequence an=1+(−1)n(n+1)/2a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }an​=1+(−1)n(n+1)/2

Find a Formula for the Nth Term of the Sequence $a _ { n } = 1 + ( - 1 ) ^ { n ( n + 1 ) / 2 }$