Statistics & Probability
Arithmetic Sequences
An arithmetic sequence has a constant difference (d) between terms. The nth term formula is: aₙ = a₁ + (n − 1)d, where a₁ is the first term.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Find the 10th term of the sequence: 3, 7, 11, 15, ...
Step-by-Step Solution
Common difference d = 7 − 3 = 4. First term a₁ = 3.
aₙ = a₁ + (n − 1)d
a₁₀ = 3 + (10 − 1)(4) = 3 + 36 = 39
Answer
39
Problem
Find the common difference and next two terms: 20, 15, 10, 5, ...
Step-by-Step Solution
d = 15 − 20 = −5 (decreasing by 5 each time)
Next terms: 5 + (−5) = 0, then 0 + (−5) = −5
Answer
d = −5; next terms are 0 and −5
Problem
Write a formula for the nth term of: 2, 5, 8, 11, ...
Step-by-Step Solution
a₁ = 2, d = 3.
aₙ = 2 + (n − 1)(3)
aₙ = 2 + 3n − 3 = 3n − 1
Answer
aₙ = 3n − 1
Problem
Is 100 a term in the sequence 4, 7, 10, 13, ...?
Step-by-Step Solution
a₁ = 4, d = 3. Formula: aₙ = 4 + (n−1)(3) = 3n + 1.
Set 3n + 1 = 100 → 3n = 99 → n = 33.
Yes! 100 is the 33rd term.
Answer
Yes, it is the 33rd term
Problem
Find the sum of the first 5 terms of: 10, 14, 18, 22, ...
Step-by-Step Solution
d = 4, a₁ = 10.
Terms: 10, 14, 18, 22, 26.
Sum = 10 + 14 + 18 + 22 + 26 = 90.
Answer
Sum = 90
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Find the 8th term: 1, 4, 7, 10, ...
What is the common difference: 50, 44, 38, 32, ...?
Find the 15th term: 5, 10, 15, 20, ...
Write nth term formula: 0, 3, 6, 9, ...
What is the 20th term of: 100, 95, 90, ...?
Find the 12th term: 2, 6, 10, 14, ...
What is the common difference: 7, 12, 17, 22, ...?
Is 50 a term in the sequence 3, 8, 13, 18, ...?
Find the 6th term: −5, −2, 1, 4, ...
Write nth term formula: 8, 5, 2, −1, ...
What is the 100th term of: 1, 2, 3, 4, ...?
Find the sum of the first 4 terms: 3, 7, 11, 15
Finished all 12? Give yourself a pat on the back — then check your work!
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