In a classroom activity, students are writing down letter patterns on the board. The first student writes "A", the second writes "AB", the third writes "ABC", the fourth writes "ABCD", and so on — each new student adds one more letter to the end of the previous sequence, always starting from A. If this process continues, which letter will appear at the 75th position overall?
Consider the following statements:
Statement I: The 75th letter falls within the sequence written by the 13th student.
Statement II: The 75th letter is 'I'.

Select the correct answer from the options below:
(a) Only Statement is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

Question

In a classroom activity, students are writing down letter patterns on the board. The first student writes "A", the second writes "AB", the third writes "ABC", the fourth writes "ABCD", and so on &mdash; each new student adds one more letter to the end of the previous sequence, always starting from A. If this process continues, which letter will appear at the 75th position overall?
Consider the following statements:
Statement I: The 75th letter falls within the sequence written by the 13th student.
Statement II: The 75th letter is 'I'.

Select the correct answer from the options below:
(a) Only Statement is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

ONLiNE UPSC · Accepted Answer

Total letters written form a sequence: 1 + 2 + 3 + ... + n = n * (n + 1)/2
We need to find smallest n such that n * (n + 1) / 2 &ge; 75
Try, n = 12:
12 &times; 13/2 = 78 &ge; 75 &rarr; So, the 75th letter is in the 12th student's sequence not 13th.
Statement I is incorrect.

Up to 11th student:
11 &times; 12/2 = 66 letters.
So, 75th letter is the 75 &ndash; 66 = 9th letter in 12th sequence = 'I'.
Statement II is correct.

Hence, option b is correct.

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In a classroom activity, students are writing down letter patterns on the board. The first student writes "A", the second writes "AB", the third writes "ABC", the fourth writes "ABCD", and so on — each new student adds one more letter to the end of the previous sequence, always starting from A. If this process continues, which letter will appear at the 75th position overall? Consider the following statements: Statement I: The 75th letter falls within the sequence written by the 13th student. Statement II: The 75th letter is 'I'. Select the correct answer from the options below: (a) Only Statement is correct. (b) Only Statement II is correct. (c) Both Statements I and II are correct. (d) None of the Statements is correct.