Problem Solving

Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

Question
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

Answer (Highlight to read)

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

The correct answer is D