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

Again a great question.........but just a suggestion....pls dont answer your question in the same post....so that we can try also we can see how other ppl's approach in solving......and at the end u can post ur answer stating the Best approach....

Thanks in advance

In the aforementioned series i.e. 1,2,4,7,11,16,22...The differences between each consecutive element is increasing by 1

2-1 = 1
4-2 = 2
7-4 = 3
11-7 = 4
16-11 = 5
22-16 = 6...So on and so forth

Note : the difference between the 2nd and 1st term is 1; between the 3rd and 2nd term is 2; between the 4th and 3rd term is 3...Hence we can conclude from this trend that the difference between the 60th and the 59th term will be 59. Thus the differential between the elements is in the form of an arithmetic progression viz 1,2,3,4,5,6.....59

The sum of an arithmetic progression = {n [ 2a + (n-1) d ] } / 2

= {59 [2*1 + (59-1) 1] } / 2
= { 59 [ 2 + 58 ] } / 2
= {59 * 60} / 2
= 59*30 = 1770

Thus the sum of all the differences is 1770 and the 1st number of the sequence is 1. Hence the answer is 1771 i.e. D

where can I read about arithmetic progression?

yvonne12 wrote:where can I read about arithmetic progression?

check out wikipedias link for arithmetic progression ... https://en.wikipedia.org/wiki/Arithmetic_progression ... post if any doubts ..