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Algebra Difficulty 4.7 AIME Find the answer

1. The following arrays are all composed of three numbers, they are:
(1,2,3),(2,4,6),(3,8,11),(4,16,20),(5,32,37),⋯ .(an,bn,cn). \begin{array}{l} (1,2,3),(2,4,6),(3,8,11), \\ (4,16,20),(5,32,37), \cdots . \\ \left(a_{n}, b_{n}, c_{n}\right) . \end{array}

Please write an expression for an,bn,cna_{n}, b_{n}, c_{n}
an= a_{n}=
,bn=b_{n}= \qquad ,cn=c_{n}= \qquad

A number or a short expression. Spacing and $ signs are ignored.

Solution

2. an=n,bn=2n,cn=n+2na_{n}=n, b_{n}=2^{n}, c_{n}=n+2^{n}.
From 1,2,3,4,5,⋯1,2,3,4,5, \cdots conjecture an=na_{n}=n;
From 2,4,8,16,32,⋯2,4,8,16,32, \cdots conjecture bn=2nb_{n}=2^{n};
From each set of numbers being “the sum of the first two equals the third” conjecture cn=n+2nc_{n}=n+2^{n}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.