Find what happens between one term and the next, then jump straight to the hundredth.
Find the 50th term
7, 11, 15, 19, …
The terms go up by 4 each time, so the rule is 4n + 3: at n = 1 that gives 7 ✓. The 50th term is 4(50) + 3 = 203. A comes from 4 × 50, forgetting the +3. B stops at the 49th term and D runs on to the 51st — both are the off-by-one that comes from counting terms instead of gaps.
The gaps tell you what kind of sequence you are looking at.
| If the gaps are… | The sequence is | Example |
|---|---|---|
| all the same | arithmetic | 7, 11, 15, 19 (+4) |
| a constant multiplier | geometric | 3, 6, 12, 24 (×2) |
| growing in a pattern | neither — describe it | 1, 4, 9, 16 (the squares) |
Check the gap between every pair, not just the first two. 2, 4, 8, 10 starts looking like doubling and is not.
nth term = dn + (first term − d), where d is the common difference
7, 11, 15, 19, …
d = 4
start with 4n
gives 4, 8, 12, 16
each is 3 short
compare with 7, 11, 15, 19
nth term = 4n + 3
add the 3 back
Always test the rule at n = 1 before trusting it: 4(1) + 3 = 7 ✓ and at n = 3: 4(3) + 3 = 15 ✓
20, 17, 14, 11, …
d = −3
−3n gives −3, −6, −9
the multiples
nth term = −3n + 23
each is 23 more
Test: −3(1) + 23 = 20 ✓ and −3(4) + 23 = 11 ✓
A falling sequence must have a negative coefficient of n. If your rule gives bigger numbers as n grows but the sequence shrinks, the sign is wrong.
nth term = first term × rn−1
3, 6, 12, 24, …
find the 7th term
3 × 26
six doublings, not seven
3 × 64 = 192
the 7th term
Find r by dividing any term by the one before it: 24 ÷ 12 = 2. Do it twice to be sure the multiplier really is constant.
Both routes agree, which is the point: the rule dn + c already has the off-by-one built into it. That is why it is worth finding.
Given a value and asked for n? Set the rule equal to it and solve. n must be a whole number.
Which term of 5, 8, 11, … is 95?
nth term = 3n + 2
d = 3; 3n gives 3, so add 2
3n + 2 = 95
set the rule equal to the value
3n = 93, n = 31
the 31st term
Check: 3(31) + 2 = 95 ✓. If n had come out as a fraction, 95 would not be in the sequence at all.
Which term of 50, 44, 38, … is the first negative one?
nth term = −6n + 56
d = −6; −6(1) + 56 = 50 ✓
−6n + 56 < 0
negative means below zero
56 < 6n, n > 9.33…
the next whole number
n = 10
the 10th term
n > 9.33 does not mean the 9th term: −6(9) + 56 = 2, still positive. The first whole number above 9.33 is 10, and −6(10) + 56 = −4 ✓.
Add the first 20 terms of 2, 5, 8, …
20th term = 3(20) − 1 = 59
rule 3n − 1
sum = (202)(first + last)
pair 1st with 20th, 2nd with 19th, …
= 10 × (2 + 59) = 610
10 pairs, each adding to 61
Every pair adds to the same 61, and 20 terms make 10 pairs. With an odd number of terms the same formula still works.
The gap method again, in less friendly settings.
Find the 9th term of 12, 54, 2, …
d = 54 − 12 = 34
54 − 24
9th term = 12 + 8(34)
8 steps after the first
= 12 + 6 = 132
Fractions do not change the gap count; write the halves as quarters to see d.
The 3rd term is 11 and the 7th term is 27. Find the rule.
27 − 11 = 16 over 4 gaps
7 − 3 = 4 steps
d = 4
16 ÷ 4
first term = 11 − 2(4) = 3
two steps back from the 3rd
nth term = 4n − 1
4(1) − 1 = 3 ✓, 4(7) − 1 = 27 ✓
Count gaps between the given positions, not the positions themselves: from 3rd to 7th is 4 steps.
Find the 6th term of 2, −6, 18, …
r = −6 ÷ 2 = −3
the multiplier
6th term = 2 × (−3)5
five multiplications
= 2 × (−243) = −486
an odd power stays negative
With a negative ratio the signs alternate: odd-numbered terms are positive, even-numbered terms negative.
Exams give a few terms and ask for the next term, the rule, a far term such as the 50th, or which term equals a given value. The fastest route is the rule dn + (first term − d); for "which term", set the rule equal to the value and solve for n. For a 10-second check, test your rule at n = 1 and n = 2 against the terms you can see.
Find the rule
A stack of chairs is 85 cm tall with one chair, and each extra chair adds 7 cm. Which rule gives the height with n chairs?
The sequence is 85, 92, 99, … with d = 7. Since 7n alone gives 7 at n = 1 and we need 85, add 78: the rule is 7n + 78. Check n = 1: 7 + 78 = 85 ✓ and n = 3: 21 + 78 = 99 ✓ A is the classic slip — it adds the first term instead of the first term minus d, and gives 92 for a single chair.
Hint: one chair is already 85 cm, so the very first term is 85, not 85 + 7. Test every candidate rule at n = 1 and three of the four die immediately.
