Number 7
2 marks · HardThe decimal number 0.345345345\ldots has a repeating block of 3 digits. What is the 50th digit after the decimal point?
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5
Common mistakes: Incorrectly finding the starting position rather than 50th digit; Giving 4 or 3 as the answer by incorrect cycle placement
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Marking points
M1 · 1 mark · Correctly divides by 3 to find the position in the cycle or shows clear identification of the repeating block.
A1 · 1 mark · Correct answer: 5 (follow-through)
Worked solution
The decimal 0.345345345... repeats the block "345" every 3 digits.
To find the 50th digit, divide 50 by 3:
So after 16 full cycles (48 digits), the 49th digit is the start of the next cycle:
49th digit: "3", 50th digit: "4", 51st digit: "5"
Therefore, the 50th digit is 4.
Correction: Since remainder is 2, the digits are:
1st: 3, 2nd: 4, 3rd: 5...
When remainder is 2, the digit is the second in "345", which is 4.