Chapter 8 of 8

All 8 chapters
  1. 1Letters for unknown numbers, readunderstood
  2. 2Putting numbers in, readunderstood
  3. 3Collecting like terms, readunderstood
  4. 4Brackets, readunderstood
  5. 5Solving equations in one step, readunderstood
  6. 6Solving equations in two steps, readunderstood
  7. 7Writing your own equations, readunderstood
  8. 8Sequences and the nth term, readunderstood

M3.8.8 Chapter 8

Sequences and the nth term

A dot pattern has 5, 8, 11 and 14 dots in its first four pictures. How many dots will the 100th picture have?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the nth term of a sequence like 5, 8, 11, 14
  • Use the nth term to find the 100th term

A sequence is a list of numbers that follows a rule. Each number in the list is a term. Here the sequence is 5, 8, 11, 14.

Draw it first

Count the dots in each picture

  1. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dots

    1 Draw the pictures and count the dots: 5, 8, 11, 14.

  2. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dotsblack: 3 more. ring: always 2

    2 Each picture adds one column of 3 black dots. The 2 ring dots never change.

  3. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dotsblack: 3 more. ring: always 2dots = 3 × picture number + 2

    3 Picture number n has 3 × n + 2 dots.

Each term has a term number. The first term has term number 1, the second has term number 2, and so on. We call the term number n.

The rule that gives any term from its term number is the nth term. For this sequence it is 3n + 2. We will see how to find it.

Look at how far each term is from the next. The terms go up by 3 each time. This is the common difference. It is the number in front of n.

A table with three columns: n, 3n and term. The rows are 1, 3, 5. Then 2, 6, 8. Then 3, 9, 11. Then 4, 12, 14.n3nterm135268391141214
Every term is 2 more than 3n.

Worked example

Find the nth term of 5, 8, 11, 14.

  1. The terms go up by 3 each time. The common difference is 3, so the nth term starts with 3n.
  2. Write 3n for each term number: 3, 6, 9, 12.
  3. Compare with the sequence: 5 − 3 = 2, 8 − 6 = 2, 11 − 9 = 2.
  4. Each term is 2 more than 3n. So the nth term is 3n + 2.
  5. Check with n = 4: 3 × 4 + 2 = 14. It matches the fourth term.

Answer: 3n + 2

Now we can find any term. We do not need to draw all the pictures. We put the term number into the nth term.

Worked example

The nth term of a sequence is 3n + 2. Find the 100th term.

  1. The 100th term has n = 100.
  2. Put n = 100 into 3n + 2. This gives 3 × 100 + 2.
  3. Multiply first: 3 × 100 = 300.
  4. Then add: 300 + 2 = 302.

Answer: 302

Four dot pictures: term 1 with 7 dots, term 2 with 11 dots, term 3 with 15 dots and term 4 with 19 dots. Each picture has 3 ring dots and columns of 4 black dots. Between the pictures, the marks +4 show the difference.term 17 dots+4term 211 dots+4term 315 dots+4term 419 dots
The sequence 7, 11, 15, 19 goes up by 4, so its nth term starts with 4n.

Worked example

Find the nth term of 7, 11, 15, 19. Then find the 50th term.

  1. The terms go up by 4 each time, so the nth term starts with 4n.
  2. The first term of 4n is 4. The first term of the sequence is 7. And 7 − 4 = 3.
  3. So the nth term is 4n + 3.
  4. For n = 50: 4 × 50 = 200.
  5. Then 200 + 3 = 203.

Answer: 4n + 3 and 203

See it three ways

With things

Put 2 stones in a ring. Add a column of 3 stones, and count all of them. Add another column, and count again.

As a picture

A number machine. The term number n = 4 goes in. The first step is × 3 and gives 12. The second step is + 2 and gives 14.4n× 312+ 214

In symbols

The nth term is 3n + 2. With n = 4 it is 3 × 4 + 2 = 14.

Try it

Grow a pattern with stones

You need: about 40 small stones or beans.

  1. Put 2 stones in a ring. Put a column of 3 stones beside it. Count them all.
  2. Add another column of 3 stones. Count them all again.
  3. Add one more column of 3. Write the three counts.
  4. Guess how many stones there are with 10 columns. Then check with 3 × 10 + 2.

What you will see: The counts are 5, 8 and 11. With 10 columns there are 3 × 10 + 2 = 32 stones. The ring of 2 is the "plus 2" in the rule.

If you cannot try it, this is what happens.
Three dot pictures: term 1 with 5 dots, term 2 with 8 dots and term 3 with 11 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference.term 15 dots+3term 28 dots+3term 311 dots

A wrong idea many people have

The nth term of 5, 8, 11, 14 is n + 3, because the terms go up by 3.

Why it feels right: The sequence adds 3 each time, so adding 3 seems to be the rule.

The test that breaks it: Put in n = 1. Then n + 3 is 4, but the first term is 5. Put in n = 4. Then n + 3 is 7, but the fourth term is 14.

What is true: The common difference is the number in front of n: it is 3n. Then fix the start: 3n + 2.

Key words

sequence
a list of numbers that follows a rule.
term number
the place of a term in the list: 1 for the first, 2 for the second.
nth term
a rule that gives any term from its term number, such as 3n + 2.
common difference
the amount a sequence goes up by each time.

Back to your guess

The answer is C: 302 dots, because each picture has 3 times its number, plus 2.

Each new picture adds a column of 3 dots, and 2 ring dots never change. Picture number n has 3n + 2 dots. For n = 100 that is 3 × 100 + 2 = 302.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The dot pictures have 4, 7, 10 and 13 dots. What is the nth term?

    Four dot pictures: term 1 with 4 dots, term 2 with 7 dots, term 3 with 10 dots and term 4 with 13 dots. Each picture has 1 ring dot and columns of 3 black dots. Between the pictures, the marks +3 show the difference.term 14 dots+3term 27 dots+3term 310 dots+3term 413 dots
    Tap your answer.

    1. Not this one. Test n = 2. Then n + 3 is 5, but the second term is 7. The difference 3 goes in front of n. Go to M3.8.8. Try another.

    2. Not this one. The terms of 4n are 4, 8, 12, 16. This goes up by 4, but the sequence goes up by 3. Go to M3.8.8. Try another.

    3. Not this one. The terms of 3n are 3, 6, 9, 12. Each term of the sequence is 1 more than that. Go to M3.8.8. Try another.

    4. Yes, that is right. The terms go up by 3, so start with 3n. The terms of 3n are 3, 6, 9, 12. Each term is 1 more. So the nth term is 3n + 1.

    See the answer

    3n + 1

    The terms go up by 3, so start with 3n. The terms of 3n are 3, 6, 9, 12. Each term is 1 more. So the nth term is 3n + 1.

    • If you chose A: Test n = 2. Then n + 3 is 5, but the second term is 7. The difference 3 goes in front of n. Go to M3.8.8.
    • If you chose B: The terms of 4n are 4, 8, 12, 16. This goes up by 4, but the sequence goes up by 3. Go to M3.8.8.
    • If you chose C: The terms of 3n are 3, 6, 9, 12. Each term of the sequence is 1 more than that. Go to M3.8.8.

  2. The nth term of a sequence is 3n + 2. What is the 50th term?

    Tap your answer.

    1. Yes, that is right. Put n = 50 into 3n + 2. 3 × 50 = 150, and 150 + 2 = 152.

    2. Not this one. You put 3 and 50 side by side as 350. But 3n means 3 × n, so 3 × 50 = 150. Go to M3.8.2. Try another.

    3. Not this one. You added 3 + 50 + 2. A number next to a letter means multiply: 3 × 50. Go to M3.8.2. Try another.

    See the answer

    152

    Put n = 50 into 3n + 2. 3 × 50 = 150, and 150 + 2 = 152.

    • If you chose B: You put 3 and 50 side by side as 350. But 3n means 3 × n, so 3 × 50 = 150. Go to M3.8.2.
    • If you chose C: You added 3 + 50 + 2. A number next to a letter means multiply: 3 × 50. Go to M3.8.2.

  3. Is 62 a term of the sequence 5, 8, 11, 14, ...?

    Tap your answer.

    1. Not this one. The list shows only the first four terms. The three dots mean it goes on for ever, so use the nth term. Go to M3.8.8. Try another.

    2. Yes, that is right. Solve 3n + 2 = 62. Take 2 from both sides: 3n = 60. Divide by 3: n = 20. It is a whole number, so 62 is the 20th term.

    3. Not this one. The rule is 3n + 2, not 3n. Take 2 from 62 first. Then 60 ÷ 3 = 20. Go to M3.8.6. Try another.

    See the answer

    Yes. 3n + 2 = 62 gives n = 20, a whole number.

    Solve 3n + 2 = 62. Take 2 from both sides: 3n = 60. Divide by 3: n = 20. It is a whole number, so 62 is the 20th term.

    • If you chose A: The list shows only the first four terms. The three dots mean it goes on for ever, so use the nth term. Go to M3.8.8.
    • If you chose C: The rule is 3n + 2, not 3n. Take 2 from 62 first. Then 60 ÷ 3 = 20. Go to M3.8.6.

  4. Find the nth term of 7, 11, 15, 19, ...

    Tap your answer.

    1. Not this one. The number added to 4n is 7 − 4 = 3, not 7. The terms of 4n are 4, 8, 12, 16, and each term is 3 more. Go to M3.8.8. Try another.

    2. Not this one. Test n = 2. Then n + 4 is 6, but the second term is 11. The difference 4 goes in front of n. Go to M3.8.8. Try another.

    3. Yes, that is right. The terms go up by 4, so start with 4n: 4, 8, 12, 16. Each term is 3 more. So the nth term is 4n + 3.

    See the answer

    4n + 3

    The terms go up by 4, so start with 4n: 4, 8, 12, 16. Each term is 3 more. So the nth term is 4n + 3.

    • If you chose A: The number added to 4n is 7 − 4 = 3, not 7. The terms of 4n are 4, 8, 12, 16, and each term is 3 more. Go to M3.8.8.
    • If you chose B: Test n = 2. Then n + 4 is 6, but the second term is 11. The difference 4 goes in front of n. Go to M3.8.8.

  5. What is the common difference of the sequence 5, 8, 11, 14?

    Tap your answer.

    1. Yes, that is right. 8 − 5 = 3, and 11 − 8 = 3, and 14 − 11 = 3. The terms go up by 3 each time.

    2. Not this one. That is the number added in 3n + 2. The difference is the step between terms. Go to M3.8.8. Try another.

    3. Not this one. That is the first term. The difference is how much the terms go up by each time. Go to M3.8.8. Try another.

    See the answer

    3

    8 − 5 = 3, and 11 − 8 = 3, and 14 − 11 = 3. The terms go up by 3 each time.

    • If you chose B: That is the number added in 3n + 2. The difference is the step between terms. Go to M3.8.8.
    • If you chose C: That is the first term. The difference is how much the terms go up by each time. Go to M3.8.8.

  6. Joseph says the nth term of 5, 8, 11, 14 is n + 3. Test it with n = 2. What do you find?

    Tap your answer.

    1. Not this one. The second term of the sequence is 8, not 5. A correct nth term must give 8. Go to M3.8.8. Try another.

    2. Yes, that is right. The right rule is 3n + 2, which gives 3 × 2 + 2 = 8 for n = 2. So n + 3 is wrong.

    3. Not this one. Put in n = 2 again. 2 + 3 is 5, not 8. Go to M3.8.2. Try another.

    See the answer

    n + 3 gives 5 for n = 2, but the second term is 8. He is wrong.

    The right rule is 3n + 2, which gives 3 × 2 + 2 = 8 for n = 2. So n + 3 is wrong.

    • If you chose A: The second term of the sequence is 8, not 5. A correct nth term must give 8. Go to M3.8.8.
    • If you chose C: Put in n = 2 again. 2 + 3 is 5, not 8. Go to M3.8.2.

  7. From an earlier chapter

    Solve 2x + 7 = 19.

    Tap your answer.

    1. Not this one. You took 7 from 19, which is right, but you stopped. This is 2x = 12. Divide by 2. Go to M3.8.6. Try another.

    2. Not this one. You added 7 to 6, or took 6 from 19. Take 7 from both sides first, then divide by 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 7 from both sides: 2x = 12. Divide by 2: x = 6. Check: 2 × 6 + 7 = 19.

    See the answer

    6

    Take 7 from both sides: 2x = 12. Divide by 2: x = 6. Check: 2 × 6 + 7 = 19.

    • If you chose A: You took 7 from 19, which is right, but you stopped. This is 2x = 12. Divide by 2. Go to M3.8.6.
    • If you chose B: You added 7 to 6, or took 6 from 19. Take 7 from both sides first, then divide by 2. Go to M3.8.6.

  8. The nth term of a sequence is 4n + 3. What is the 20th term?

    See the answer

    83

    Put n = 20 into 4n + 3. 4 × 20 = 80, and 80 + 3 = 83.

  9. What is the common difference of the sequence 6, 11, 16, 21?

    See the answer

    5

    11 − 6 = 5, and 16 − 11 = 5, and 21 − 16 = 5. The terms go up by 5 each time.

Keep it

Copy this into your exercise book.

Find the common difference. It is the number in front of n. Compare with that many n, and add what is left. Then put in n = 100 to jump far ahead.

Draw: Write 5, 8, 11, 14 and under it 3, 6, 9, 12. Write 2 more under each. Write the nth term 3n + 2.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (850 KB). It opens on any phone.