Arithmetic sequences and series

The sequence starts at 5 and goes up by 3. The green line is u₁ + (n − 1)d; the red one is the version people write from memory, u₁ + nd. Slide n and watch where each one lands.

n = 1

5u₁ + (n−1)d

8u₁ + nd

3 outalways by d

At n = 1 the correct formula gives the first term. The other one has already skipped it.

The two formulae

u_n = u₁ + (n − 1)d, to reach the nth term you take n − 1 steps, because you are already standing on the first one.

S_n = n2(2u₁ + (n − 1)d), or equivalently n2(u₁ + u_n), which is quicker when you already know the last term.

u₁ = 5, d = 3 5, 8, 11, 14, … u₁₀= 5 + 9(3) = 32 S₁₀= 5(2(5) + 9(3)) = 5(37) = 185 check→ 102(5 + 32) = 5(37) = 185 ✓

Working backwards

Most questions give you terms and want d, or give you a term and want n. Both are one line of algebra.

u₃ = 11, u₇ = 23→ 4 steps cover 12, so d = 3 u_n = 62→ 5 + 3(n − 1) = 62 → n − 1 = 19, so n = 20

n must come out a positive whole number. If it does not, either the term is not in the sequence or something upstream is wrong. That is a free check on every backwards question, and it catches the missing minus one immediately.

When the model is not quite arithmetic

Real data rarely steps by exactly the same amount. An arithmetic model is still useful, and the mark is for saying how it departs: whether the gaps are drifting in one direction, whether a single year breaks the pattern, and whether you would trust it ten terms out. Saying “it is approximately arithmetic” and stopping is the answer that scores least.

Worked example

A stadium has 24 seats in the front row, and each row behind has 4 more than the one in front. There are 30 rows.

How many seats are in the back row, and how many in the whole stand?

u₁ = 24, d = 4, n = 30Write these down first. Half the errors in this sub-topic are reading the wrong number as the first term.
u₃₀ = 24 + (30 − 1) × 4Twenty-nine steps, not thirty, because row 1 is where you start.
= 24 + 116 = 140 seatsCheck it: thirty rows gaining 4 each should land somewhere near 24 + 120, and it does.
S₃₀ = ³⁰⁄₂ (24 + 140)The first-plus-last form, because the last term is already known.
= 15 × 164 = 2460 seatsSanity check: 30 rows averaging 82 seats gives 2460. The average of the first and last row is exactly the average row, which is why the formula works.

Your turn

1. u₁ = 5 and d = 3. Find u₂₀.

2. For the same sequence, find S₂₀.

3. u₃ = 11 and u₇ = 23. Find d.

Where the marks go

Stating u₁ and d before anything else. Half the errors in this sub-topic are reading the wrong term as the first one.

Using (n − 1), and checking it by putting n = 1 back in. If the formula does not return the first term, it is wrong.

Choosing the right sum formula. If the question already gives the last term, the n/2(first + last) version is one step shorter and harder to get wrong.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

A sequence begins 3, 6, 12, 24. Find the 10th term.

Geometric, not arithmetic. The ratio is 2, not a common difference, so uₙ = 3 × 2⁹ = 1536. Always test both: subtract consecutive terms, then divide them.
Which method?

A loan's balance after k monthly payments falls, but not in a straight line. Is that an arithmetic sequence?

No. The amount coming off the debt changes every month, so the differences are not equal. It is geometric in the balance. Equal differences are what make a sequence arithmetic.
Which method?

Write 0.00046 in standard form.

Standard form: 4.6 × 10⁻⁴. a must be at least 1 and under 10, which is why it is not 0.46 × 10⁻³.

The same idea elsewhere

1.3, geometric sequences: The same n − 1 appears in the geometric formula, in the power instead of the multiplier.
4.7, discrete random variables: Summing a sequence and finding E(X) are both weighted totals; the sigma notation is shared.

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