Arithmetic or geometric in context

In 7 of 20 recent exams
1% of the marks
Paper 1 or 2
5 to 14 marks

How it is set in the exam

Paper 2 (calculator) more than Paper 1 (no calculator), 5–14 marks spread over several parts, often a Section B question. This type is the set-up; later parts usually turn into Find n or Compound interest and depreciation.

Spot it in the exam

Exam phrases to spot (4)

“each row has 3 more seats than the row in front”
“increases by 4% each year”
“in the 10th year”
“the total”

In disguise (6)

no mention of a sequence at all: stacked cans, a training plan, a salary offer, sales figures
the starting value is "now" or "at the start of 2015" rather than "year 1", so the powers shift by one
calendar years: "in 2030", when 2021 is the first year
a running total "at the start of year 10", which covers only 9 years
two plans to compare, one arithmetic and one geometric
simple interest: the same interest every year, so arithmetic with $d$ = the yearly interest

Looks like it, but isn't (1)

yearly values given as data in a table, with "the regression line": a straight line fitted by the GDC, Regression on the GDC. A rate given in words ("increases by 4% each year") is a sequence; a table of recorded values is regression. One question can ask for both and compare them (that type's Example 1.4)

Worked example

Where counting starts: "at the start of 2015"

Paper 2 (calculator)

Question

At the start of 2015 a town has $24\,000$ people. The population decreases by $3\%$ each year, and it is recorded at the start of each year. Find the first year in which the recorded population is below $20\,000$ . [4]

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Steps

Make the three decisions in Reading a context: arithmetic or geometric; $u_1$ and what it means; a term or a sum.
Pin down where counting starts. Write which $n$ the question's year (or row) is: "now" is time $0$ (power $n$ ); "year 1" or "2021" is $u_1$ (power $n - 1$ ).
Write the expression with the right power or number of terms, and calculate.
Answer in context: whole rows, money to the stated accuracy, a calendar year. If $n$ is the unknown, it is now Find n.

Common mistakes

Off by one year
a salary in "year 10" is $u_1 r^9$ . A value 10 years after "now" is (the value now) $\times\, r^{10}$ . Decide whether the first value is $u_1$ or time $0$ before you write a power.
Calendar years
if 2021 is $u_1$ , then 2030 is $u_{10}$ , and the answer $n = 17$ is the year 2037, not 2038.
"At the start of year 10" is $S_9$
, not $S_{10}$ .

More mistakes (4)

Term when a total is asked
(or the reverse). Underline "total", "altogether", "in the $n$ th", "in 2030".
Using $r = 0.03$ for "3% increase".
The multiplier is $1.03$ .
Treating simple interest as compound.
€1500 at 6% simple interest is worth $1500 + 90n$ after $n$ years, not $1500 \times 1.06^n$ .
Unrealistic answers
$12.4$ rows or $1049.6$ seats. Answer in the units of the context.

Easy to confuse with

Next typeThree consecutive terms with an unknown

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Easy to confuse with

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