Practice · Grade 10 Sequences & Series
Arithmetic Sequence — Practice
Training problems on the nth term and the sum in rising difficulty, plus one boss question. Hint and full working for every task. Grade 10, free.
Q1 of 7
0 correct
Find the nth term of the sequence.
a₁ = 4, d = 3, n = 10 → a₁₀
Quick answer
What is the best way to practice arithmetic sequences?
Work several problems in rising difficulty: first the nth term with aₙ = a₁ + (n−1)·d, then the sum with Sₙ = n/2 · (2a₁ + (n−1)d). Deliberately practice falling sequences (negative d), write down every substitution step, and check your result via the average of the first and last term. Reveal a hint one time too often rather than too few — understanding beats fast guessing.
HowTo
A 4-step solving strategy
This order works for every arithmetic sequence — nth term or sum.
- 1Step 1 of 4
Read off a₁, d and n
Identify the first term a₁, the common difference d = a₂ − a₁ and the target n. Ask yourself: is the nth term or the sum being asked for?
- 2Step 2 of 4
Pick the right formula
For a single term: aₙ = a₁ + (n−1)·d. For the sum of the first n terms: Sₙ = n/2 · (2a₁ + (n−1)d).
- 3Step 3 of 4
Substitute the values — with signs
Plug a₁, d and n into the formula. Watch negative d: (n−1)·d can be negative, e.g. 11·(−2) = −22.
- 4Step 4 of 4
Compute and check
Evaluate the bracket first, then the rest. Sanity check for the sum: Sₙ = n/2 · (a₁ + aₙ) — average of first and last term times n.
Examples
Worked practice examples with full steps
Four typical problem types. Try each one yourself first, then compare with the working.
Easy
nth term: a₁ = 3, d = 5, n = 10
aₙ = a₁ + (n−1)·d
a₁₀ = 3 + (10−1)·5
= 3 + 9·5 = 3 + 45
a₁₀ = 48 ✓ (sequence 3, 8, 13, …)
Standard form. Add the difference (n−1) times to the first term.
Easy
Falling sequence: a₁ = 20, d = −3, n = 7
a₇ = 20 + (7−1)·(−3)
= 20 + 6·(−3)
= 20 − 18
a₇ = 2 ✓
Negative d: the sequence falls. (n−1)·d becomes negative, so subtract.
Medium
Sum: a₁ = 2, d = 4, n = 8
Sₙ = n/2 · (2a₁ + (n−1)d)
S₈ = 8/2 · (2·2 + 7·4)
= 4 · (4 + 28) = 4 · 32
S₈ = 128 ✓
Evaluate the bracket first, then multiply by n/2 = 4.
Hard
Gauss sum: 1 + 2 + … + 100
a₁ = 1, d = 1, n = 100
S₁₀₀ = 100/2 · (1 + 100)
= 50 · 101
S₁₀₀ = 5050 ✓
The famous Gauss sum — pairing first + last term.
Pitfalls
Common mistakes — and how to avoid them
These traps show up in almost every test.
Confusing (n−1) with n
The nth term adds (n−1)·d, not n·d. The first term (n=1) gets zero steps: a₁ = a₁ + 0·d.
Computing the difference the wrong way
d = a₂ − a₁, that is later minus earlier. For falling sequences d is negative — e.g. 10, 8, 6, … with d = −2.
Dropping the factor n/2 in the sum
Sₙ = n/2 · (2a₁ + (n−1)d). Computing only the bracket gives twice (or a multiple of) the correct result.
Sign error with negative d
With d = −2 and n = 12, (n−1)·d = 11·(−2) = −22 — not +22. Carry the minus sign through.
Confusing arithmetic with geometric
Arithmetic sequences add d; geometric ones multiply by a ratio q. Linear vs. exponential growth.
Study
Practice with a plan — three quick tips
15 minutes at a time, not 90 at once
Three short sessions on three days stick better than one long session the night before the test. The keyword is spaced repetition.
Solve first, then look at the solution
Write down your working before revealing the hint. Active recall is far more effective for learning than passive reading.
For every wrong answer: why?
Was it (n−1) instead of n? A sign with negative d? The missing n/2? Note the cause — next time you will spot the mistake instantly.
FAQ
Frequently asked questions about practicing
Glossary
Terms in one sentence
- Arithmetic sequence
- A number sequence with a constant difference between consecutive terms.
- Term aₙ
- A single value of the sequence; the nth one is aₙ.
- First term a₁
- The starting value of the sequence.
- Common difference d
- The constant step between two terms: d = a₂ − a₁.
- Partial sum Sₙ
- The sum of the first n terms of the sequence.
- Gauss sum
- The pairing idea behind Sₙ = n/2 · (a₁ + aₙ).
- Boss question
- The last and hardest task of a practice set.