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All 11 Flashcards — nth term
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Question
What is an arithmetic sequence?
Answer
A sequence where each term differs from the previous one by a constant amount, the common difference d. Example: 4, 7, 10, 13 has d = 3.
Question
What is the common difference, and how do you find it?
Answer
The constant gap between consecutive terms: d = uₙ − uₙ₋₁. Subtract any term from the next. Example: in 9, 5, 1 the difference is d = −4.
Question
An arithmetic question asks for 'the value of u₂₀' in one part and 'which term equals 100' in another — how do you tell them apart?
Answer
Both use uₙ = u₁ + (n − 1)d. Want a VALUE (u₂₀)? Put n = 20 and compute. Want a POSITION ('which term = 100')? Set the formula equal to that value and solve for n. Spot value-vs-position first.
Question
Why is it (n − 1)d and not nd in the nth-term formula?
Answer
You start at u₁ and add d only on each step after the first, so reaching the nth term takes (n − 1) steps. Example: u₅ = u₁ + 4d.
Question
How do you find d from two terms, e.g. u₃ = 17 and u₇ = 41?
Answer
Divide the difference by the number of steps between them: (41 − 17) ÷ (7 − 3) = 24 ÷ 4 = 6.
Question
How do you find which term equals a given value?
Answer
Set uₙ = u₁ + (n − 1)d equal to the value and solve for n. Example: 4 + (n−1)5 = 99 ⇒ n = 20.
Question
Given a rule like uₙ = 20 − 4n, how do you read u₁ and d?
Answer
Substitute n = 1 for the first term (u₁ = 16); the coefficient of n is the common difference (d = −4).
Question
When are three terms u₁, u₂, u₃ arithmetic?
Answer
When the differences are equal: u₂ − u₁ = u₃ − u₂. The middle term is the average of its neighbours.
Question
How do you find an unknown k so that k+2, 2k+3, 5k−2 are arithmetic?
Answer
Set u₂ − u₁ = u₃ − u₂ and solve: (k + 1) = (3k − 5) ⇒ k = 3.
Question
What is the difference between a sequence and a series?
Answer
A sequence is the list of terms (3, 7, 11, …); a series is their sum (3 + 7 + 11 + …).
Question
In a word problem, do you add d n times or (n − 1) times?
Answer
Spot u₁ first. If you want a TERM ('the 6th row'), it's n − 1 jumps (row 1 = 0 jumps). If u₁ is a starting amount and you want the value 'after n years/steps', it's n jumps. Always ask: how many times do I add d to get there?
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Arithmetic sequences & series
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