Find n in a sequence

Find $n$ : which term, how many terms, least $n$

In 16 of 20 recent exams
2% of the marks
Paper 1 or 2
2 to 6 marks

How it is set in the exam

Paper 1 (no calculator) or Paper 2 (calculator), 2–6 marks, often the last part of a context question

Spot it in the exam

First, split

" $=$ value" (which row has 50 seats? how many rows hold 1050 seats?) → an equation, and $n$ must come out a whole number. " $>$ value", "exceeds", "less than" → an inequality, and you round in the direction the question needs: least $n$ → up; greatest $n$ / the most that fit → down.

Exam phrases to spot (6)

“find which term…”
“how many terms…”
“find the least value of $n$ for which…”
“the smallest number of terms”
“in which year does … first exceed…”
“the greatest number of rows that can be built”

In disguise (9)

"how many terms must be added for the total to exceed…"
"the first term that is negative" (asks for $n$ or for the term's value: read which)
"the most rows that fit", "how many terms are less than 500": a greatest $n$ , so round down
"the maximum value of $S_n$ " for a decreasing arithmetic sequence: the sum peaks at the last positive term, where the terms reach zero
"within 0.001 of the sum to infinity": $S_\infty - S_n < 0.001$
"after how many complete years is it worth less than half…", "how long until the investment doubles" (compound interest with the time unknown)
"the first month in which Ben's savings overtake Anna's"
rows of equal shapes stacked until they pass a given height: each row adds the same height, such as the height of an equilateral triangle of side $s$ , $\frac{\sqrt{3}}{2}s$ (Right-angled triangles and exact values), so the least number of rows comes from an inequality in $n$
the least $n$ for which a running total fills whole boxes with none spare (say $S_n$ divisible by $12$ ): no equation or inequality gives it, so list $S_n$ in a table and take the first $n$ that works

A sigma with an unknown top limit (see Sigma notation) is solved the same way, but it stays in Sigma notation, because decoding the sigma is the hard part.

Looks like it, but isn't (6)

"show that the total never reaches…": no $n$ works, so nothing is found for $n$ . The proof is the sum to infinity: Sum to infinity, and when it exists
an exponential equation with nothing to count (no terms, periods or whole years), such as $2^{x+1} = 7$ : Solve an exponential equation (unknown in the power, or in the base)
a formula in continuous time, such as $N_0 \times 1.035^t$ or $500e^{-0.04t}$ , even when a whole number of units or a calendar year is asked: Exponential and log models (solve for $t$ , then round to the unit asked
the number of trials is fixed and the probability of at least one success is asked: $1 - \mathrm{P}(\text{none})$ , with no unknown $n$ (Tree diagrams and repeated draws)
the least number of trials for at least $k \ge 2$ successes, each trial with the same probability: no formula to solve, so tabulate $\mathrm{P}(X \ge k) = 1 - \mathrm{P}(X \le k - 1)$ against $n$ : Binomial probabilities: recognise B(n, p), then translate the words (for at least one, $1 - (1 - p)^n$ stays here)
a least number of games, items or tickets for an expected total to reach a target: Expected value in context: fair games, totals and the least number. The inequality is linear, $n\,\mathrm{E}(X) - \text{fixed cost} \ge \text{target}$ , and the round-up rule here finishes it

Worked example

Equations: which row, how many rows

Paper 1 (no calculator)

Question

The first row of a theatre has $18$ seats, and each row has $2$ more seats than the row in front of it.
(a) Find which row has $50$ seats. [2]
(b) The theatre has $1050$ seats in total. Find the number of rows. [4]

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See it move: Which row, how many rows

$S_n = 800$ gives $n = 21.03$ , not a whole number. $S_{21} = 798 < 800$ and $S_{22} = 858 > 800$ : the least $n$ with $S_n > 800$ is 22, and the most rows with at most 800 seats is 21.
Each bar is a row: row $n$ has $u_n = 18 + 2(n - 1)$ seats, and the first $n$ rows hold $S_n = \frac{n}{2}(36 + 2(n - 1)) = n^{2} + 17n$ seats. A total is $S_n$ , not $u_n$ .
(b) 1050 seats in total: $n(n + 17) = 1050$ , so $(n + 42)(n - 25) = 0$ . $n = -42$ is impossible for a number of rows: 25 rows. Check: $u_{25} = 66$ and $\frac{25}{2}(18 + 66) = 1050$ .
" $=$ a value" is an equation, and $n$ must come out whole. "Exceeds", "more than" or "the most that fit" is an inequality: a least $n$ rounds up, a greatest $n$ rounds down, and you show the whole numbers either side, one short and the next over.

Which method?

arithmetic term
linear inequality
same
arithmetic sum, or two arithmetic sums compared
quadratic inequality, using $n > 0$
table, or solve and round (step 4)
geometric term or sum
isolate $r^n$ and compare with powers ( $2^{10} = 1024$ , $3^5 = 243$ ); logs if asked, flipping the sign for $0 < r < 1$
table, or solve and round (step 4)
arithmetic against geometric
not set
table only

What must pass the value
arithmetic term	linear inequality	same
arithmetic sum, or two arithmetic sums compared	quadratic inequality, using $n > 0$	table, or solve and round (step 4)
geometric term or sum	isolate $r^n$ and compare with powers ( $2^{10} = 1024$ , $3^5 = 243$ ); logs if asked, flipping the sign for $0 < r < 1$	table, or solve and round (step 4)
arithmetic against geometric	not set	table only

An equation (" $=$ value") is solved directly: linear for an arithmetic term, a quadratic for an arithmetic sum (reject the negative root), powers for geometric.

Steps

Write the quantity in $n$ : $u_n$ , $S_n$ , $S_\infty - S_n$ , or the difference of two sums.
Write the equation or inequality with the correct sign ("exceeds" is strict, $>$ ). Solving the equation ( $=$ ) instead of the inequality is accepted; an inequality the wrong way round loses the set-up mark ( in some markschemes, in others).
Solve by the method in the table.
For an inequality, round in the direction the question needs: least $n$ → up; greatest $n$ / the most that fit → down. Then show the two whole numbers either side: one fails and the next works (for a least $n$ ), or one works and the next fails (for a greatest $n$ ).
Answer what was asked: $n$ , the term's value, or the calendar year.

Common mistakes

Not reversing the inequality
after dividing by $\ln r$ with $0 < r < 1$ .
Rounding the wrong way.
A least $n$ rounds up: $n > 27.6$ means $n = 28$ . A greatest $n$ rounds down: $n < 125.25$ means $n = 125$ . Ask which one the question wants before rounding.
An inequality the wrong way round
writing $S_n < 120$ for "exceeds 120" loses the set-up mark ( in some markschemes, in others), even if the final $n$ happens to be right. Writing the equation $S_n = 120$ instead is accepted for that mark.

More mistakes (6)

Missing the strict inequality
if $S_{10} = 120$ exactly, then "exceeds 120" needs $n = 11$ .
"Never reaches" shown with a few sums
calculate $S_\infty$ and compare it with the target.
$n$ against $n - 1$ in the power
$u_n$ uses $r^{n-1}$ , and $S_n$ uses $r^n$ .
Giving the answer as a decimal
( $n = 29.65$ ). It is a count, so give a whole number.
$S_\infty - u_n$ instead of $S_\infty - S_n$
it scores no method mark.
Giving $n$ when the question asks for the value or the year.
"Find the first negative term" wants the term itself. "In which year" wants a calendar year: if 2021 is $u_1$ , then $u_n$ is the year $2020 + n$ , not $2021 + n$ .

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