Manhattan Prep
\(n\) is an integer greater than or equal to \(0\). The sequence \(t_n\) for \(n>0 \) is defined as \(t_n=t_{n-1}+n\). Given that \( t_0=3 \), is \(t_n\) even?
1) \(n + 1\) is divisible by 3.
2) \(n - 1\) is divisible by 4.
OA B
\(n\) is an integer greater than or equal to \(0\).
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t_0 = 3
t_1 = 3 + 1
t_2 = 3 + 1 + 2
t_3 = 3 + 1 +2 + 3
t_4 = 3 + 1 + 2 + 3 + 4
and so on, so t_n is just equal to 3 plus the sum of the first n positive integers. The sum of the first n positive integers is just (n)(n+1)/2 (it's the number of terms, n, times the average term, which, since it's an equally spaced list, is the average of the first and last terms, so is (n+1)/2 ). So we know:
t_n = 3 + (n)(n+1)/2
So t_n will be even only when (n)(n+1)/2 is even.
Statement 1 alone isn't much help - n might be 5 or 8, and then (n)(n+1)/2 will be odd or even, respectively. But Statement 2 is sufficient, because if n-1 is divisible by 4, then n is odd, and n+1 is only divisible by a single 2, so (n+1)/2 is odd. Since n and (n+1)/2 are both odd, so is (n) * (n+1)/2. So 3 + (n)(n+1)/2 will necessarily be a sum of two odd numbers, and thus is even.
t_1 = 3 + 1
t_2 = 3 + 1 + 2
t_3 = 3 + 1 +2 + 3
t_4 = 3 + 1 + 2 + 3 + 4
and so on, so t_n is just equal to 3 plus the sum of the first n positive integers. The sum of the first n positive integers is just (n)(n+1)/2 (it's the number of terms, n, times the average term, which, since it's an equally spaced list, is the average of the first and last terms, so is (n+1)/2 ). So we know:
t_n = 3 + (n)(n+1)/2
So t_n will be even only when (n)(n+1)/2 is even.
Statement 1 alone isn't much help - n might be 5 or 8, and then (n)(n+1)/2 will be odd or even, respectively. But Statement 2 is sufficient, because if n-1 is divisible by 4, then n is odd, and n+1 is only divisible by a single 2, so (n+1)/2 is odd. Since n and (n+1)/2 are both odd, so is (n) * (n+1)/2. So 3 + (n)(n+1)/2 will necessarily be a sum of two odd numbers, and thus is even.
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