HSPT: Number series: add the previous two terms
Learn to continue a series where each term is the sum of the two before it, and to fill a gap by working backward.
How do you continue a series where each term comes from the two terms before it?
Short answer
Add the last two terms to get the next one. To find a missing term, subtract a known term from the sum it helped make.
Know first: Adding whole numbers, Constant-difference series
Step 1 of 5
Each new term is the sum of the two before it
In a sum-of-two series, every term after the first two is made by adding the two terms just before it. The gaps don't follow a steady rule, so the subtraction tests from other series won't help.
The question often states the rule. Even when it doesn't, you can spot it: check whether each term equals the two before it added together.
Checking the rule on 2, 3, 5, 8, 13
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
Every term checks out, so the next term is 8 + 13 = 21.
Step 2 of 5
Add the last two terms
Worked example: A sum-of-two series
Each term after the first two is the sum of the two before it. What comes next? 3, 7, 10, 17, 27, 44, …
- A71
- B88
- C61
- D115
- 1
Check the rule
3 + 7 = 10, 7 + 10 = 17, 10 + 17 = 27, 17 + 27 = 44. It holds every time.
- 2
Add the last two terms
27 + 44 = 71.
- 3
See why the other choices are there
88 doubles 44. 61 adds the last gap, 17, to 44. 115 goes one term too far: it is 44 + 71, the term after next.
Answer
71
Find the wrong step
A student continues this sum-of-two series: 5, 12, 17, 29, 46, …
- 1
Checks the rule: 5 + 12 = 17, 12 + 17 = 29 and 17 + 29 = 46.
- 2
Adds the last term to itself: 46 + 46 = 92.
What went wrong
The rule adds the last two different terms. Doubling uses only one of them.
- 3
Chooses 92.
The fix
Add the last two terms: 29 + 46 = 75.
Step 3 of 5
Add forward, or subtract to go back
Check yourself · Question 1
Each term after the first two is the sum of the two before it. What comes next? 2, 9, 11, 20, 31, 51, …
Answer: B
The last two terms are 31 and 51, so the next term is 31 + 51 = 82.
Trap answer: 102
Doubling the last term
- Why it looks right
- Doubling is fast, and on big numbers it feels close to adding two terms.
- Why it's wrong
- The rule says the two preceding terms. 51 + 51 uses only one.
- The right answer
- 82, because 31 + 51 = 82.
Check yourself · Question 2
Each term after the first two is the sum of the two before it. What number belongs in the blank? 6, ___, 13, 20, 33
Answer: A
The third term, 13, equals 6 plus the blank, so the blank is 13 − 6 = 7. Check: 7 + 13 = 20, and 13 + 20 = 33.
Common mistake
"I just double the last number, since that is about the same as adding the last two."
- Why it's tempting
- In a sum-of-two series the last two terms can look close in size.
- Do this instead
- Write the sum out with both numbers, like 31 + 51. The rule always uses two different terms.
Step 4 of 5
What to remember
Remember
- Check that each term equals the two before it added together.
- The next term is the last two terms added, never the last term doubled.
- To find a missing term, subtract a known term from the sum it helped make.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which two terms did you add in the practice question, and which wrong choice comes from doubling?