Find t_{20} given t_1 = 3 and d = 4.
1
Recall the formula
t_n = t_1 + (n-1)d
2
Substitute
t_{20} = 3 + (20-1)(4)
3
Simplify
t_{20} = 3 + 76
Math 10 · Lesson 1
Arithmetic
Sequences.
Every step the same size. One formula, any term. Watch the pattern click into place.
3
7
11
15
19
What is an arithmetic sequence?
INTERACTIVE
The Arithmetic Detector
Drop in 4 numbers. We will check the differences and call it.
INTERACTIVE
Sequence Builder
1
2
1t1
3t2
5t3
7t4
9t5
11t6
13t7
15t8
tn = 1 + (n − 1) · 2 = 2n − 1
One formula. Any term.
DERIVATION
Where does (n − 1) come from?
1
add d once to reach the 2nd term
2
add d twice for the 3rd term
3
three steps to the 4th term
4
count the gaps, not the terms
CALCULATOR
General Term Calculator
t10 = 28
From easy to tricky.
In an arithmetic sequence, t_5 = 17 and t_{10} = 32. Find t_1 and d.
1
Two unknowns, two equations
t_1 + 4d = 17 and t_1 + 9d = 32
2
Subtract
5d = 15, so d = 3
3
Solve for t_1
t_1 + 4(3) = 17, so t_1 = 5
How many terms are in 11, 5, -1, -7, ..., -175?
1
Identify
t_1 = 11, d = -6, t_n = -175
2
Solve for n
-175 = 11 + (n-1)(-6)
3
Isolate
-186 = -6(n-1), so n - 1 = 31
Show that x - 1, 3x + 2, 5x + 5 forms an arithmetic sequence for any value of x.
1
Compute t_2 - t_1
(3x + 2) - (x - 1) = 2x + 3
2
Compute t_3 - t_2
(5x + 5) - (3x + 2) = 2x + 3
3
Conclude
Both differences equal 2x + 3, so the sequence is arithmetic for every x.
It is always the same process.
Every arithmetic-sequence problem you will ever see reduces to the same four steps. Learn them once.
STEP 1
Identify
List what you know (t₁, d, n, t_n) and what you need to find. Count the unknowns.
STEP 2
Equations
Set up exactly as many equations as you have unknowns. One unknown ⇒ one equation. Two unknowns ⇒ two.
STEP 3
Solve
For one equation, isolate the unknown. For two, subtract to cancel t₁ and solve for d first.
STEP 4
Verify
Plug your answer back into the formula. The original facts must come out true.
Unknowns = equations you need
| Unknowns | Equations |
|---|---|
| One (e.g., find t₂₀ given t₁ and d) | 1 |
| Two (e.g., t₁ and d unknown; two terms given) | 2 |
| Three (rare; needs three terms) | 3 |
The shortcuts pros use.
d is just a slope
Plot the sequence as (n, t_n) and every arithmetic sequence is a straight line. The common difference d is the slope. Spotting d visually is often faster than computing.
EX → t_n = 3n + 2 → slope 3 → d = 3
Subtract to cancel t₁
Two terms given, t₁ and d unknown? Write both equations and subtract. The t₁ cancels and you have a single equation in d.
EX → (t_1 + 9d) − (t_1 + 4d) = 5d
"Between" problems hide n
Inserting numbers between two given values? Count the resulting terms. Three numbers between A and B means 5 terms total: A + 4d = B.
EX → Insert 3 between −5 and 11 ⇒ 5 terms ⇒ d = 4
Coefficient of n is d
For any general term written as t_n = an + b, the constant a is d, and t₁ = a + b. Read both off the formula without computing terms.
EX → t_n = 7n + 5 → d = 7, t_1 = 12
Negative steps still count
When t_n decreases, d is negative — but n is still a positive integer ≥ 1. Find-n problems with d < 0 work the same; just track signs carefully.
EX → d = −6 from 11 to −175 ⇒ n = 32
Verify, always
After solving for t₁ and d, plug back into both given facts. If both come out true, you are done. If not, recheck your subtraction.
EX → t_1 = 5, d = 3 ⇒ t_5 = 17 ✓ t_{10} = 32 ✓
Test yourself.
The Test
29 problems across three tiers. Auto-graded on submit. Hints available before you submit, full solutions after.
0/29Answered
Core
Core (10)
C1
Find the next term: 4, 7, 10, 13, ...
Need a hint?
- What is the common difference d?
- d = 3. Add it to the last term.
C2
What is the common difference d for 100, 95, 90, 85, ...?
Need a hint?
- d = t_2 − t_1.
C3
Find t_{10} for the sequence 3, 10, 17, 24, ...
Need a hint?
- Use t_n = t_1 + (n − 1)d with t_1 = 3, d = 7.
- t_{10} = 3 + 9(7).
C4
Is the sequence 5, 8, 12, 17 arithmetic?
Need a hint?
- Check that all consecutive differences are equal.
C5
Write the next three terms of 2.5, 2, 1.5, ...
Need a hint?
- d = −0.5.
C6
Find t_{15} given t_1 = −5 and d = 4.
Need a hint?
- t_{15} = −5 + 14(4).
C7
Find d if t_1 = 7 and t_5 = 27.
Need a hint?
- t_5 = t_1 + 4d.
- 27 = 7 + 4d, so 4d = 20.
C8
Find t_1 if t_{10} = 50 and d = 5.
Need a hint?
- 50 = t_1 + 9(5).
C9
Write the general term t_n for the sequence 1, 4, 7, 10, ...
Need a hint?
- Find t_1 and d, then plug into t_n = t_1 + (n − 1)d.
- t_1 = 1, d = 3, so t_n = 1 + (n − 1)(3).
C10
If t_n = 3n + 2, what is t_7?
Need a hint?
- Substitute n = 7.
Apply
Apply (10)
A1
Insert three numbers between −5 and 11 to form an arithmetic sequence.
Need a hint?
- With three numbers inserted, the sequence has 5 terms total.
- −5 + 4d = 11, so d = 4.
A2
Fill in the blanks to create an arithmetic sequence: ___, ___, 8, ___, ___, −7
Need a hint?
- 8 is the 3rd term and −7 is the 6th term.
- 8 + 3d = −7, so d = −5.
A3
How many terms are in the sequence 11, 5, −1, −7, ..., −175?
Need a hint?
- t_1 = 11, d = −6, t_n = −175. Solve t_n = t_1 + (n − 1)d for n.
- −175 = 11 − 6(n − 1).
A4
An arithmetic sequence has t_5 = 28 and t_{12} = 7. Write the general term.
Need a hint?
- Set up two equations and subtract.
- 7d = −21, so d = −3.
A5
Using t_n = 43 − 3n (from apply-4), how many terms are larger than −30?
Need a hint?
- Solve 43 − 3n > −30.
- 73 > 3n, so n < 24.33.
A6
Show that x − 1, 3x + 2, 5x + 5 is an arithmetic sequence.
Need a hint?
- Compute t_2 − t_1 and t_3 − t_2 and check they are equal.
A7
Determine the position of 97 in the sequence −3, 1, 5, 9, ...
Need a hint?
- t_n = −3 + (n − 1)(4). Solve t_n = 97.
A8
5x + 2, 7x − 4, and 10x + 6 are consecutive arithmetic terms. Find x.
Need a hint?
- Consecutive differences are equal: (7x − 4) − (5x + 2) = (10x + 6) − (7x − 4).
A9
If x, y, z are the first three terms of an arithmetic sequence, express z in terms of x and y.
Need a hint?
- Equal differences: y − x = z − y.
A10
Wolf Creek Golf Course tee-off times begin at 8:00 and are 8 minutes apart. Treat 8:00 as time 0 (minutes). Write the general term t_n for the n-th tee-off time in minutes.
Need a hint?
- First tee-off is at minute 0; each next one is +8 minutes.
Extension
Extension (9)
E1
Susan starts a fitness program. On day 1 she does 1 push-up, day 2 she does 3, day 3 she does 5, and so on. (a) Write the general term t_n for the number of push-ups on day n. (b) On what day will she do 101 push-ups?
Need a hint?
- Identify t_1 and d, then build t_n.
- For (b), solve 2n − 1 = 101.
E2
In alkanes (a family of hydrocarbons), the number of hydrogen atoms relates to the number of carbon atoms: 1C/4H (methane), 2C/6H (ethane), 3C/8H (propane), 4C/10H (butane). (a) Write a formula for the number of hydrogen atoms H given C carbon atoms. (b) How many hydrogen atoms are in an alkane with 100 carbon atoms?
Need a hint?
- Treat carbon count as n and hydrogen count as t_n.
- d = 2, t_1 = 4 (when C = 1).
E3
Atmospheric pressure at sea level is 14.7 psi. Water pressure adds another 14.7 psi for every 30 feet (≈10 m) of depth. (a) Write the pressures at depths corresponding to n = 1, 2, 3, 4 (where n is the count of 30-ft increments). (b) Write the general term. (c) What is the pressure at 1000 ft? (d) At 2000 ft?
Need a hint?
- Each 30 ft adds 14.7 psi.
- 1000 ft is 1000/30 ≈ 33.3 increments — but the answer key treats t_n = 14.7n where n is depth in 30-ft units, giving t_n at any n. Plug in n = 1000/30 = 33.33.
E4
A swimmer trains for an event. Her first session lasts 4 minutes, and each subsequent session adds 4 more minutes. (a) Write the first five terms. (b) Write the general term. (c) After how many sessions will she swim for 320 minutes?
Need a hint?
- Solve t_n = 320 with t_n = 4n.
E5
The number of registered beekeepers in a province decreased steadily. In year 1 there were 850, and the count decreased by the same amount each year, reaching 502 in year 13. What is the annual change d?
Need a hint?
- Use t_1 = 850, t_{13} = 502, and t_n = t_1 + (n − 1)d.
E6
A bus departs every 5 minutes starting at 13:54. (a) Write the first five departure times. (b) Treating minutes since midnight, write the general term t_n for the n-th departure. (c) What time is the 24th departure?
Need a hint?
- 13:54 in minutes since midnight is 834. Each step adds 5.
E7
For an arithmetic sequence with first term t_1 and common difference d, which statement is FALSE?
Need a hint?
- When d > 0, the terms always increase, regardless of t_1's sign.
E8
If the general term of an arithmetic sequence is t_n = 7n + 5, identify t_1, the common difference d, and write the first three terms.
Need a hint?
- Compute t_1, t_2, t_3 by substituting n = 1, 2, 3.
E9
When the terms of an arithmetic sequence are plotted as (n, t_n), the resulting graph is:
Need a hint?
- The general term t_n = t_1 + (n − 1)d is linear in n.