Find u₁ and d (or r)

Find $u_1$ and $d$ (or $r$ ) from two facts

In 11 of 20 recent exams
2% of the marks
Paper 1 or 2
3 to 8 marks

How it is set in the exam

Paper 1 (no calculator) or Paper 2 (calculator), 3–8 marks. One of the two most common sequences questions (with Find n), usually part (a), followed by "hence find $u_{20}$ ", "find $S_{15}$ " or a "show that".

Spot it in the exam

Exam phrases to spot (3)

“the third term is 7 and the tenth term is 35”
“find the first term and the common difference”
“find the common ratio”

In disguise (6)

one fact is a sum: "the sum of the first 10 terms is 185", or a sum to infinity
a relation instead of a value: "the fifth term is three times the second", " $u_5 + u_7 = 22$ " (one equation, not two)
two geometric terms an even number of steps apart: $r^2 = 9$ gives two sequences, and a later detail ("the fifth term is negative", "the terms alternate in sign") picks one
the two facts link an arithmetic and a geometric sequence (e.g. $u_2 = v_2$ , $u_4 = v_3$ )
the facts are logs: $u_1 = \ln 8$ and $u_5 = \ln 128$ give $4d = \ln 16$ , so $d = \ln 2$
one unknown trapped inside a sum formula on , which only the GDC can solve. If the condition is an inequality and the unknown is a whole number, the GDC gives the boundary value: take the whole number on the side where the inequality holds, and check its neighbours as in Find n

Looks like it, but isn't (3)

"the 2nd, 5th and 11th terms of an arithmetic sequence form a geometric sequence" is one condition on three terms: Three consecutive terms.
the facts only set the sequence up, and the question asks for $n$ (which term has a given value, or the least $n$ ): Find n, with this type as its first step
the two facts are a sector's perimeter and area: Perimeter and area both given: find r and θ (the same structure: an equation each, eliminate, reject a root

Worked example

Two terms, then a sum

Paper 1 (no calculator)

Question

In an arithmetic sequence, $u_4 = 17$ and $u_{11} = 52$ .
(a) Find the common difference and the first term. [5]
(b) Find the sum of the first 15 terms. [2]

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See it move: Two facts pin the sequence down

Two unknowns, two facts: $u_1 + 3d = 17$ and $u_1 + 10d = 52$ . Subtract: $7d = 35$ , so $d = 5$ and $u_1 = 17 - 3(5) = 2$ .
$u_{4}$ is $u_1 + 3d$ , not $u_1 + 4d$ , and from $u_{4}$ to $u_{11}$ is $11 - 4 = 7$ steps: the staircase has 7 risers, each $d$ high.
(b) $S_{20} = \frac{20}{2}\left(2(2) + 19(5)\right) = 10 \times 99 = 990$ : the $2u_1 + (n - 1)d$ form, since $u_{20}$ wasn't given.
A sum as the second fact is still linear in $u_1$ and $d$ : "the sum of the first ten terms is 185" means $\frac{10}{2}(2u_1 + 9d) = 185$ , not $u_1 + 9d = 185$ . Divide by 5 before combining.

Steps

Write each fact as an equation with the formula ( $u_n$ , $S_n$ or $S_\infty$ ). Two unknowns need two equations.
Both arithmetic:
the equations are linear in $u_1$ and $d$ . Subtract to remove $u_1$ . On : GDC: solve simultaneous linear equations.
Both geometric:
divide one equation by the other to cancel $u_1$ . You get $r^k =$ number.
If $k$ is even and $r^k > 0$ , $r$ has two values ( $\pm$ ); if $k$ is odd, one. Keep both unless something in the question rules one out.
One fact of each type
(linked sequences): make $d$ the subject and substitute. A quadratic in $r$ follows, and $r = 1$ usually has to be rejected (it makes $d = 0$ ).
Substitute back to find $u_1$ , then answer what was actually asked (a later term, a sum, or a "show that").

Common mistakes

Reading $S_{10}$ as $u_{10}$
"the sum of the first ten terms is 185" means $\frac{10}{2}(2u_1 + 9d) = 185$ , not $u_1 + 9d = 185$ .
Premature rounding
in Example 1.4, using $r = 1.09$ gives $u_{10} \approx 434$ , not $425$ . Keep the full value in the calculator (keep full-accuracy values).
Losing a root
$r^2 = 9$ gives $r = \pm 3$ . Only reject one if the question gives a reason.

More mistakes (3)

Subtracting geometric equations.
Geometric facts are products, so divide them. Subtracting $u_1 r^3 - u_1 r$ gets you nowhere.
Using $n$ instead of $n - 1$
$u_4 = u_1 + 3d$ , not $u_1 + 4d$ .
Keeping $r = 1$ for linked sequences
it makes $d = 0$ , so both sequences are constant. Reject it and say why.

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