Chapter 22 · Consecutive Natural Numbers

Build Consecutive Sequences with a Middle Term

第22讲 · 连续自然数

Represent consecutive natural, even, odd, and prime sequences precisely. Use the first term, middle term, average, product, and structural conditions to reconstruct every term.

One anchor term fixes the whole sequence.

10 interactive missionsoriginal examples + transfer challengesAutosaves in this browser

Your answers save automatically in this browser.

Learn from this page

Read a worked example, predict what the lab will show, then test your prediction. Finish a number entry and press Tab (or choose another control) to update a lab. The local hints and explanations work without opening another website.

Useful words and symbols

Consecutive means following in order. Consecutive integers have gap 1; consecutive even or odd numbers have gap 2. Positive integers start at 1. A lab explicitly allowing zero includes 0 as well.

Remainder is what is left after making equal groups. “17 mod 5 = 2” and “17 ≡ 2 (mod 5)” both say that 17 leaves remainder 2 when divided by 5. A residue class groups numbers with the same remainder.

To prove an answer is the best possible, explain why a better one cannot work, then give an example that reaches your bound.

Mission 1

Know which kind of “consecutive” you are building

The chapter introduction groups consecutive natural numbers, consecutive even numbers, consecutive odd numbers, and consecutive primes under one broad theme—but their gaps are not all the same.

Not complete

Natural numbersa, a+1, a+2, …Fixed gap: 1

Even numbersa, a+2, a+4, …Fixed gap: 2; every term even

Odd numbersa, a+2, a+4, …Fixed gap: 2; every term odd

Prime numbers2, 3, 5, 7, 11, …Neighboring primes; gaps vary

Sequence generator

Sequence type First term Number of terms Expected gap

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

Gap between consecutive odd numbersOnly even primeGap between 18 and next evenAre prime gaps always equal?

Mission 2

Use the middle term and average

For an odd number of equally spaced terms, the middle term is also the average. Terms equally far from the middle form equal-sum pairs.

Not complete

Odd number of terms

sum = number of terms × middle term

For three consecutive natural numbers:

a−1, a, a+1 → sum = 3a

Even number of terms

average = mean of the two middle terms

For four consecutive natural numbers:

a, a+1, a+2, a+3

average = (a+1+a+2)÷2 = a+1.5

Middle-term builder (zero is allowed; negative terms are not valid)

Sequence typeMiddle termOdd number of termsCommon gap

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

Middle of 38, 39, 40Sum of 7 evens centered at 12Average of 23, 25, 27Middle of 60,62,64,66,68

Mission 3

Rebuild Worked example 2

Three consecutive natural numbers have a sum divisible by 13. Their greatest term leaves remainder 4 when divided by 9. Find the least possible triple.

Not complete

Example 2

Let the middle number be a

a−1, a, a+1

The sum is 3a. Because 13 and 3 share no factor, divisibility of 3a by 13 forces:

a ≡ 0 (mod 13)

The greatest term leaves remainder 4 when divided by 9:

a+1 ≡ 4 (mod 9) → a ≡ 3 (mod 9)

Candidate ledger

The table shows multiples of 13 until the first one also has remainder 3 modulo 9.

Middle a	Multiple of 13?	a mod 9	Both conditions?	Triple

Complete the worked example result

Least middle numberSmallest termGreatest termSum of the triple

Mission 4

Recover a sequence from its total

If the count and common gap are known, the total determines the first term. The page verifies that the reconstructed first term is a whole number and that every term has the required parity.

Not complete

Consecutive-sum solver

Total sumNumber of termsCommon gapParity requirement

original Exercise 6

Five consecutive even numbers total 320.

middle = 320÷5 = 64

60, 62, 64, 66, 68

Test 22 transfer

Twenty consecutive positive integers total 290.

average = 290÷20 = 14.5

5, 6, …, 23, 24

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

Largest of four naturals totaling 54Smallest of five evens totaling 320First of twenty positives totaling 290Last of those twenty numbers

Mission 5

Reconstruct consecutive numbers from a product

Estimate the size of the factors, then test consecutive starts systematically. Multiply back to prove the reconstruction.

Not complete

Consecutive-product search

Target productNumber of consecutive factorsLargest start to testSequence gap

Exercise 2

6×7×8×9 = 3024

The four students have consecutive ages measured in completed whole years. The oldest student is the last term.

Exercise 3

5×6×7×8 = 1680

The requested value is the first term.

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

Oldest of four consecutive whole-number ages whose product is 3024Smallest of four consecutive positive whole-number factors whose product is 1680Average (midpoint) of 6,7,8,9Product of 9,11,13,15,17

Mission 6

Find consecutive composite runs and consecutive prime blocks

“Consecutive composite” means adjacent natural numbers that are all composite. “Consecutive prime” means neighboring entries in the prime list.

Not complete

First composite run

Run lengthSearch through

Consecutive prime block

Starting position in the prime list (position 1 is 2)

Seven odd primes have an odd total. Therefore a seven-prime block with an even total must include the only even prime, 2.

Exercise 1

The first four consecutive composite numbers

24, 25, 26, 27

Their sum is 102. The next example extends the run by one more composite number, 28, giving a minimum five-number sum of 130.

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

First term of the four-composite runSum of the first four-composite runSum of first five-composite runFourth term of seven consecutive primes, listed greatest to least, whose sum is even

Mission 7

Use extremes, ratios, and range averages

A consecutive sequence can also be pinned down by a relationship between its first and last terms, or by the average of a complete arithmetic range.

Not complete

Extreme-ratio solver

Number of termsCommon gapLast = multiplier × first

Average of multiples in a range (both endpoints included)

DivisorLower boundUpper bound

Test 22 · Question 7

Seven consecutive even numbers

The last is three times the first:

a+12=3a → a=6

6,8,10,12,14,16,18

Test 22 · Question 9

Multiples of 3 from 1 through 100

The first is 3 and the last is 99, so the average is:

(3+99)÷2 = 51

Checkpoint — use the fixed numbers in each question, even if you changed the explorer

Largest in the seven-even sequenceIts middle termAverage of multiples of 3 from 1–100How many such multiples?

Mission 8

Test lower bounds and complete constructions

An inequality gives a lower bound. The next allowed parity value must then be tested by building the full consecutive sequence.

Not complete

Least first term for a required minimum sum

Number of termsCommon gapMinimum sumParity of the first term only

Test 22 · Question 8

Three consecutive odd numbers

The least total not below 70 is:

23+25+27=75

Verify the previous candidate fails

21+23+25=69<70

This proves minimality.

Complete proof pattern

1. Create a lower bound.
2. Adjust to the required parity.
3. Build the sequence.
4. Reject the previous candidate.

Transfer checkpoint

Smallest term of three consecutive positive odd numbers with the least sum of at least 70Middle term of that three-odd-number sequenceSum of that sequenceSum of the preceding odd triple (each term is 2 smaller)

Try it on your own — a new problem

This extra practice has its own saved checkpoint. Try it, check your answer, then use a hint or the separate worked review.

Five consecutive even numbers total 170. What is the largest?

Your answer

Explain your reasoning: Why would a gap of 1 fail the “all even” requirement?

Mission 9

Consecutive-sequence workshop

Correct all eight questions to complete this workshop. All responses are numerical.

Not complete

1. Largest of four consecutive naturals totaling 54

2. Least middle of three consecutive positive whole numbers whose sum is divisible by 13 and whose greatest term leaves remainder 4 on division by 9

3. Smallest of five consecutive evens totaling 320

4. Oldest age when four consecutive ages multiply to 3024

5. Smallest sum of four consecutive positive integers that are all composite

6. Fourth term of seven consecutive primes, listed greatest to least, whose sum is even

7. Average of multiples of 3 from 1 through 100

8. Largest of twenty consecutive positives totaling 290

Workshop score0 / 8

Mission 10

Exit ticket and certificate

Correct all five questions and complete the other nine missions to unlock the certificate.

Not complete

1. First term of twenty consecutive positives totaling 290

2. Largest of seven evens when largest = 3 × smallest

3. Least possible first term of three consecutive positive odd numbers whose sum is at least 70

4. Smallest of four consecutive naturals with product 1680

5. Smallest sum of five consecutive positive integers that are all composite

Exit score0 / 5

Optional reflection — Which anchor was most useful: first term, middle term, average, product, or parity?

Certificate of mastery

Consecutive-Sequence Architect

Awarded to

Student

for building, reconstructing, and proving consecutive natural, even, odd, and prime sequences.

Lesson 22.1 · Chapter 22

original and lesson-design note

Interactive solvers, complete searches, parity checks, and assessments are added instructional scaffolds.Consecutive primes are treated as neighboring primes, not as a fixed-gap arithmetic sequence.