Guess the missing number

Level pending

What's the missing number in the following sequence?

2, 6, 30, 210, ?, 30030, 510510, ...


The answer is 2310.

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1 solution

Chew-Seong Cheong
Dec 28, 2015

Let a term before 2 2 be a 0 = 1 a_0 = 1 , then the following terms for n 1 n \ge 1 are given by a n = a n 1 p ( n ) a_n = a_{n-1}p(n) where p ( n ) p(n) is the n t h n^{th} prime number. Therefore, we have:

a 1 = a 0 × p ( 1 ) = 1 × 2 = 2 a 2 = a 1 × p ( 2 ) = 2 × 3 = 6 a 3 = a 2 × p ( 3 ) = 6 × 5 = 30 a 4 = a 3 × p ( 4 ) = 30 × 7 = 210 a 5 = a 4 × p ( 5 ) = 210 × 11 = 2310 a 6 = a 5 × p ( 6 ) = 2310 × 13 = 30030 a 7 = a 6 × p ( 7 ) = 30030 × 17 = 510510 \begin{array} {lllr} a_1 & = a_0 \times p(1) = 1 \times 2 & = & 2 \\ a_2 & = a_1 \times p(2) = 2 \times 3 & = & 6 \\ a_3 & = a_2 \times p(3) = 6 \times 5 & = & 30 \\ a_4 & = a_3 \times p(4) = 30 \times 7 & =& 210 \\ a_5 & = a_4 \times p(5) = 210 \times 11 & = & \boxed{2310} \\ a_6 & = a_5 \times p(6) = 2310 \times 13 & = & 30030 \\ a_7 & = a_6 \times p(7) = 30030 \times 17 & = & 510510 \end{array}

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