Arithmetic sequences & series
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What is an arithmetic sequence?
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All Flashcards in Topic 1.2
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1.2.111 cards
What is an arithmetic sequence?
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.
What is the common difference, and how do you find it?
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.
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?
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.
Why is it (n − 1)d and not nd in the nth-term formula?
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.
How do you find d from two terms, e.g. u₃ = 17 and u₇ = 41?
Divide the difference by the number of steps between them: (41 − 17) ÷ (7 − 3) = 24 ÷ 4 = 6.
How do you find which term equals a given value?
Set uₙ = u₁ + (n − 1)d equal to the value and solve for n. Example: 4 + (n−1)5 = 99 ⇒ n = 20.
Given a rule like uₙ = 20 − 4n, how do you read u₁ and d?
Substitute n = 1 for the first term (u₁ = 16); the coefficient of n is the common difference (d = −4).
When are three terms u₁, u₂, u₃ arithmetic?
When the differences are equal: u₂ − u₁ = u₃ − u₂. The middle term is the average of its neighbours.
How do you find an unknown k so that k+2, 2k+3, 5k−2 are arithmetic?
Set u₂ − u₁ = u₃ − u₂ and solve: (k + 1) = (3k − 5) ⇒ k = 3.
What is the difference between a sequence and a series?
A sequence is the list of terms (3, 7, 11, …); a series is their sum (3 + 7 + 11 + …).
In a word problem, do you add d n times or (n − 1) times?
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?
1.2.211 cards
You are given u₁ = 7 and d = 4 and asked for the sum of the first 20 terms. What do you reach for — and what is the time-trap?
Go straight to Sₙ = (n/2)(2u₁ + (n − 1)d): S₂₀ = 10(14 + 19×4) = 900. Trap: do not waste time finding u₂₀ first — the u₁-and-d form needs only what you are given.
A question asks 'how many terms until the running total first passes 500?'. How do you set it up?
It is a TOTAL, so use the sum: set Sₙ > 500 and solve for n, then round UP to the next whole number (on Paper 2, scan the GDC table of Sₙ). Spot 'total/altogether' ⇒ Sₙ, not uₙ.
How do you choose which sum formula to use?
Know u₁ and d → use (n/2)(2u₁ + (n − 1)d). Know u₁ and the last term → use (n/2)(u₁ + uₙ).
If you are told Sₙ as a formula, how do you find the first term?
u₁ = S₁ — substitute n = 1 into the sum. Example: Sₙ = 2n² + 3n ⇒ u₁ = 5.
How do you recover any term from a sum formula Sₙ?
uₙ = Sₙ − Sₙ₋₁ — the running total up to n minus the running total up to n − 1.
How can you tell a sequence is arithmetic from its sum?
Its sum is a quadratic in n with no constant term (Sₙ = an² + bn). The common difference is 2a.
In an arithmetic sequence u₅ = 20 and S₅ = 70. How do you find u₁?
Use S₅ = (5/2)(u₁ + u₅): 70 = (5/2)(u₁ + 20) ⇒ u₁ + 20 = 28 ⇒ u₁ = 8.
Why is there a factor of n/2 in the sum formula?
Pairing the first and last terms gives a constant total u₁ + uₙ, and there are n/2 such pairs, so Sₙ = (n/2)(u₁ + uₙ).
How do you find d when given two sums, e.g. S₅ and S₆?
u₆ = S₆ − S₅ gives a term; combined with the sum formula you can solve for u₁ and d.
Find S₈ for u₁ = 10, u₈ = 45.
S₈ = (8/2)(10 + 45) = 4 × 55 = 220.
What is the difference between Sₙ and uₙ?
uₙ is a single term (the nth one); Sₙ is the total of the first n terms: Sₙ = u₁ + u₂ + … + uₙ.
1.2.38 cards
What does the sigma symbol Σ mean?
Add up. Substitute the index from the lower limit to the upper limit into the expression and sum the results. Example: Σ r=1→4 of (2r+1) = 3+5+7+9 = 24.
In sigma notation, what are the lower and upper limits?
The lower limit (below Σ) is where the index starts; the upper limit (above Σ) is where it stops. Both endpoints are included.
How many terms are in a sigma sum?
Upper limit − lower limit + 1. Example: Σ r=2→9 has 9 − 2 + 1 = 8 terms.
How do you know a sigma sum is an arithmetic series?
When the summand is linear in the index (like 3r + 2). The common difference equals the coefficient of the index.
How do you evaluate Σ r=1→10 of (3r + 2) by hand?
First term 5, last term 32, n = 10; then S = (10/2)(5 + 32) = 185.
For a sum of a linear term, how do you read u₁ and d?
u₁ = the summand at the lower limit; d = the coefficient of the index. Then use the arithmetic sum formula.
What is the most common sigma mistake?
Counting terms as (upper − lower) and forgetting the +1, or assuming the index starts at 1.
How can you evaluate a sigma sum on the GDC?
On Paper 2 use sum(seq(expression, index, lower, upper)). On Paper 1 you must use the arithmetic sum formula by hand.
1.2.49 cards
How do you spot an arithmetic model in a word problem?
Look for a quantity that changes by the same amount each step (a fixed raise, a fixed number per row). Then u₁ = the start and d = the constant change.
How do you translate 'starts at 20, rises by 4 each time'?
u₁ = 20 and d = 4. The nth value is uₙ = 20 + (n − 1)4.
In a decreasing arithmetic sequence, when is the sum Sₙ greatest?
At the last term that is still positive or zero — find where uₙ = 0. Adding later negative terms only shrinks the total.
How do you find the maximum sum of an arithmetic sequence?
Solve uₙ = 0 for n, then evaluate Sₙ at that position. Example: u₁ = 48, d = −3 ⇒ u₁₇ = 0 ⇒ S₁₇ = 408.
How can the GDC help find a maximum sum (Paper 2)?
Graph Sₙ or scan a table of Sₙ and read off the largest value; the peak is at the term where uₙ = 0.
How do you find the first term past a threshold?
Set up an inequality with uₙ, solve for n, then round to the next whole number (n must be a positive integer).
A sequence has u₁ = 90, d = −7. Which is the first term below 20?
90 − 7(n − 1) < 20 ⇒ n > 11 ⇒ n = 12; u₁₂ = 13.
Does 'total' mean uₙ or Sₙ?
A total or 'altogether' is a sum, so use Sₙ. A single 'nth' value is a term uₙ.
Why must n be a whole number in application problems?
n counts terms (rows, years, balls), which only come in whole numbers; round a decimal n to the appropriate integer and check.
Topic 1.2 study notes
Full notes & explanations for Arithmetic sequences & series
Math AA SL exam skills
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