TACHS: Arithmetic sequences: the first term plus the steps
Find any term of an adding pattern, find which term has a given value, and work with two known terms, figure differences and short sums.
How do you find a term far down a sequence that adds the same number each time?
Short answer
Find the common difference, count the steps from the first term (one fewer than the term number), and compute first term + steps × difference.
Know first: Multiplying whole numbers, Finding the difference between two numbers
Step 1 of 6
Count steps, not terms
In 4, 10, 16, 22, … each term is 6 more than the one before. That 6 is the common difference.
To reach the 15th term you do not add 6 fifteen times. The 1st term is already there, so you take 15 − 1 = 14 steps. That gives the rule: nth term = first term + (n − 1) × difference.
| Term number | Steps from term 1 | Value |
|---|---|---|
| 1 | 0 | 4 |
| 2 | 1 | 4 + 6 = 10 |
| 4 | 3 | 4 + 3 × 6 = 22 |
| 15 | 14 | 4 + 14 × 6 = 88 |
The number of steps is always one less than the term number.
Step 2 of 6
Use the rule, then check a term you can see
Worked example: The 15th term
A sequence begins 4, 10, 16, 22, … and keeps adding the same number. What is the 15th term?
- A84
- B88
- C90
- D94
- 1
Find the difference
10 − 4 = 6, and 16 − 10 = 6.
- 2
Count the steps
15 − 1 = 14 steps from the 1st term.
- 3
Compute
4 + 14 × 6 = 4 + 84 = 88.
- 4
Check the rule on a known term
The 4th term should be 4 + 3 × 6 = 22. It is.
- 5
See why the other choices are there
84 is 14 × 6 without the starting 4. 90 is 15 × 6, which leaves out the start and takes one step too many. 94 adds 15 steps instead of 14.
Answer
The 15th term is 88.
Which term equals 70?
In the sequence 4, 10, 16, 22, …, which term equals 70?
- 1
Steps needed: (70 − 4) ÷ 6 = 66 ÷ 6 = 11.
- 2
So 70 is the 11th term.
What went wrong
11 is the number of steps after the 1st term. The term number is one more than the steps.
The fix
Eleven steps after the 1st term is the 12th term. Check: 4 + 11 × 6 = 70.
Step 3 of 6
Two known terms, figure differences and sums
If you know two terms, the steps between them give the difference. When the 3rd term is 14 and the 8th term is 39, there are 8 − 3 = 5 steps covering 39 − 14 = 25, so the difference is 5. Go back 2 steps from the 3rd term to get the first term: 14 − 10 = 4.
For how many more one figure has than another, the start cancels. If each figure adds 4 tiles, Figure 11 has (11 − 5) × 4 = 24 more tiles than Figure 5.
A short sum by pairing
3 + 7 + 11 + 15 + 19 + 23
Pair first and last: 3 + 23 = 26, 7 + 19 = 26, 11 + 15 = 26
Three pairs: 3 × 26 = 78
Step 4 of 6
Steps are one fewer than terms
Check yourself · Question 1
A sequence begins 2, 9, 16, 23, … and continues by adding the same number. What is its 20th term?
Answer: B
The difference is 7. The 20th term is 19 steps after the 1st: 2 + 19 × 7 = 2 + 133 = 135.
Trap answer: 142
One step too many
- Why it looks right
- The question says 20th term, so multiplying the difference by 20 matches the number you see.
- Why it's wrong
- The 1st term already counts as one of the twenty. Only 19 steps separate it from the 20th.
- The right answer
- 135, because 2 + 19 × 7 = 135.
Check yourself · Question 2
The 5th term of an arithmetic sequence is 23 and the 9th term is 39. What is the first term?
Answer: C
From the 5th to the 9th term is 4 steps covering 39 − 23 = 16, so the difference is 4. The 1st term is 4 steps before the 5th: 23 − 4 × 4 = 7. Check: 7 + 8 × 4 = 39.
Check yourself · Question 3
A dot pattern has 6 dots in Figure 1, and each figure has 5 more dots than the one before. How many more dots does Figure 12 have than Figure 7?
Answer: A
From Figure 7 to Figure 12 is 12 − 7 = 5 steps, and each step adds 5 dots: 5 × 5 = 25. Check: Figure 12 has 6 + 11 × 5 = 61 and Figure 7 has 6 + 6 × 5 = 36, and 61 − 36 = 25.
Common mistake
"The 20th term is 20 times the difference."
- Why it's tempting
- The term number and the difference are the only numbers that seem to matter.
- Do this instead
- Start from the first term and take n − 1 steps: first term + (n − 1) × difference. Test your rule on the 2nd term, which should be one step from the start.
Step 5 of 6
What to remember
Remember
- nth term = first term + (n − 1) × difference.
- To find a term's position, find the steps with (value − first term) ÷ difference, then add 1.
- Between two terms or figures, the difference is the number of steps × the common difference.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, how many steps separate the first term from the one you want, and which wrong choice takes one step too many?