1
Sum the first 200 terms of a periodic sequence
Not answeredConsider the sequence:
3213576213576…
The first term is 3. After it, the block 2, 1, 3, 5, 7, 6 repeats forever.
Find the sum of the first 200 terms.
total
Lesson 21.7 — Chapter 21 Exercise
A complete English student edition of Test 21. Work through periodic sequences, exact and modular digit-sum counts, complete decimal blocks, carry structure, extremal-number construction, number–digit-sum equations, and verified constructions.
Exercise directions
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Tips for this exercise
Digit-sum notations(n) means the sum of the decimal digits of n. For example, s(2012)=2+0+1+2. A natural number on this page means a positive integer.
Numerals and distinct digitsab means the two-digit numeral 10a+b. “All digits are different” means no written digit may repeat, and a numeral never begins with zero.
Give every requested resultQuestion 12 asks for every possible box digit. Question 14 accepts any correctly verified pair. Questions 11–14 also require written explanations.
Section I · Fill-in questions
Questions 1–10 · 6 points each · 60 points
2
Count three-digit numbers with digit sum 25
Not answeredHow many three-digit positive integers have digit sum:
hundreds digit + tens digit + ones digit = 25?
numbers
Optional case table
3
Total digit sum from 1 through 1000
Not answeredFind the total of all decimal digits appearing in the positive integers from 1 through 1000, inclusive. In symbols, find:
s(1)+s(2)+···+s(1000).
Optional complete-block calculation
4
Use a two-digit addition and its carry
Not answeredLet ab and cd be two-digit numerals:
ab + cd = 149.
Find a+b+c+d.
Numeral conventionab means 10a+b, not a×b. Because the numerals are two-digit, a and c are nonzero digits.
Optional column-addition work
5
Build the smallest odd number with digit sum 44
Not answeredAn odd positive integer N has digit sum 44. Find the least possible value of N.
Optional place-value argument
6
Count numbers with distinct digits totaling 7
Not answeredHow many positive integers have all of the following properties?
- Their digit sum is 7.
- No decimal digit is repeated.
Numbers may have different lengths, but a leading zero is never written.
numbers
Optional count by numeral length
7
Construct the least odd and even numbers with digit sum 22
Not answeredFind both:
Least odd number
Its digit sum is 22.
Its digit sum is 22.
Least even number
Its digit sum is 22.
Its digit sum is 22.
Optional construction argument
8
Count three-digit numbers with digit sum 10
Not answeredHow many three-digit positive integers have digit sum 10?
numbers
Optional case count
9
Find the rank of 2012 in a filtered list
Not answeredThe positive integers satisfying both conditions below are arranged in increasing order:
- the integer is divisible by 4;
- its digit sum is 5.
Counting the first qualifying number as position 1, what is the position of 2012 in that ordered list? You only need to consider qualifying integers up to and including 2012.
th value
Optional list or counting argument
10
Recover the total of four distinct digits
Not answeredThe digits a,b,c,d are all different, and:
ab + cd = 182.
Find a+b+c+d.
Numeral conventionab and cd are two-digit numerals, so a and c are nonzero.
Optional carry and distinct-digit reasoning
Section II · Extended-response questions
Questions 11–14 · 14 points each · 56 points
11
Count digit sums divisible by 7 among the first 200 integers
Not answeredAmong the positive integers from 1 through 200, how many have a digit sum divisible by 7?
numbers
Optional practice hint
The digit sum is positive and at most 19 in this range (199 has digit sum 19). Which multiples of 7 can it equal? Count those cases separately, splitting into 1–99, 100–199, and 200.
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12
Find every possible digit in a number–digit-sum difference
Not answeredA four-digit positive integer N is reduced by the sum of its own digits. The result is the four-digit number:
N − s(N) = 603□.
Find every possible digit represented by the box and give at least one valid value of N for each possibility.
ClarificationThe printed question asks for “the digit,” but the stated conditions allow more than one value. This student edition asks for every mathematically possible digit instead of silently selecting one.
Optional practice hint
Write N using thousands, hundreds, tens, and ones. Subtract its digit sum and factor out 9. Which values 603□ can be multiples of 9? A divisibility check is necessary; also construct a four-digit witness for each ending.
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13
Solve a number–digit-sum equation
Not answeredA three-digit positive integer equals 18 times the sum of its digits:
N = 18s(N).
Find the three-digit integer.
Optional practice hint
A three-digit number has digit sum at most 27. If its digit sum is S, the proposed number is 18S. Use divisibility by 9 to reduce the possible values of S, then check the digits of every remaining candidate.
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14
Construct one lucky pair
Not answeredTwo integers form a lucky pair when:
- their difference is 2;
- the digit sum of each integer is divisible by 7.
Find one lucky pair whose two integers both lie from 100 through 200, inclusive.
Construction noteMore than one pair may satisfy the conditions. Any correctly verified pair earns full credit.
Optional practice hint
Adding 2 usually raises the digit sum by 2, but a carry changes that. Look near the end of a tens block. Verify the gap, the range, and both digit sums for your chosen pair.
Submit your worksheet to open the help available for this attempt.