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GMAT/Problem Solving - Sequence Sixty
Sequence Sixty
What is the sixtieth term in the following sequence?
1, 2, 4, 7, 11, 16, 22...
(A) 1,671
(B) 1,760
(C) 1,761
(D) 1,771
(E) 1,821
ОТВЕТЫ И КОММЕНТАРИИ
Решение
Noting that a1 = 1, each subsequent term can be calculated as follows: a1 = 1 a2 = a1 + 1
a3 = a1 + 1 + 2
a4 = a1 + 1 + 2 + 3 ai = a1 + 1 + 2 + 3 + ... + i-1
In other words, ai = a1 plus the sum of the first i - 1 positive integers. In order to determine the sum of the first i - 1 positive integers, find the sum of the first and last terms, which would be 1 and i - 1 respectively, plus the sum of the second and penultimate terms, and so on, while working towards the median of the set. Note that the sum of each pair is always equal to i:
1 + (i - 1) = i
2 + (i – 2) = i
3 + (i – 3) = i
…
Because there are (i - 1)/2 such pairs in a set of i - 1 consecutive integers, this operation can be summarized by the formula i(i - 1)/2. For this problem, substituting a1 = 1 and using this formula for the sum of the first (i-1) integers yields: ai = 1 + (i)(i - 1)/2
The sixtieth term can be calculated as: a60 = 1 + (59)(60)/2 a60 = 1,771
Cubic Conundrum
You have a six-sided cube and six cans of paint, each a different color. You
may not mix colors of paint. How many distinct ways can you paint the cube
using a different color for each side? (If you can reorient a cube to look like
another cube, then the two cubes are not distinct.)
Вверх 
