Three consecutive terms with an unknown

In 6 of 20 recent exams
1% of the marks
Paper 1, no calculator
4 to 7 marks

How it is set in the exam

Paper 1 (no calculator), 4–7 marks, often with a "show that" and a rejected value. Sometimes a long Section B question that tests the same three terms as arithmetic in one part and as geometric in another.

Spot it in the exam

Exam phrases to spot (2)

“ $x$ , $x + 6$ and $5x - 6$ are the first three terms of a geometric sequence”
“are consecutive terms of an arithmetic sequence. Find $x$ .”

In disguise (6)

terms of one sequence form another: "the 2nd, 5th and 11th terms of an arithmetic sequence are the first three terms of a geometric sequence". Write those three terms in $a$ and $d$ , then it is the same question.
the same three expressions, two conditions: "in this part, the sequence is arithmetic … in this part, the sequence is geometric". Each part starts again from the expressions.
binomial coefficients: "the coefficients of $x$ , $x^2$ and $x^4$ in the expansion … are consecutive terms of a geometric sequence". The binomial theorem gives the three terms (Expand, or find one term or coefficient builds them); the rest is this type.
"multiplying by the same number takes $x$ to $6$ and $6$ to $x + 5$ " (equal ratios, never saying "geometric")
"find the possible values of the common ratio" when the terms contain $k$
log terms: $\ln A$ , $\ln B$ , $\ln C$ in arithmetic sequence means $2\ln B = \ln A + \ln C$ , so $B^2 = AC$

Looks like it, but isn't

an arithmetic and a geometric sequence linked by two separate facts ( $u_2 = v_2$ , $u_4 = v_3$ ). That is Find $u_1$ and $d$ (or $r$ ) from two facts.

Worked example

Geometric: a quadratic, one root rejected

Paper 1 (no calculator)

Question

The first three terms of a geometric sequence are $x$ , $x + 6$ and $5x - 6$ , where $x > 0$ .
(a) Find the value of $x$ . [5]
(b) Find the sixth term of the sequence. [2]

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Steps

1
Write the condition on the three terms $A$ , $B$ , $C$ (if they come from somewhere else, such as an expansion or another sequence, write them out first):
- arithmetic: equal differences, $B - A = C - B$ (so $2B = A + C$ )
- geometric: equal ratios, $\frac{B}{A} = \frac{C}{B}$ (so $B^2 = AC$ )
Solve by rearranging and factorising; never divide by $x$ or $d$ . The geometric condition usually gives a quadratic, so expect two values.
Test each value against the question: a stated sign ( $x > 0$ ), $d \neq 0$ , a zero term (it forces the middle term to be 0, and then a ratio is $\frac00$ ), $\lvert r \rvert < 1$ if a sum to infinity is mentioned, and positive log arguments.
Then find what is asked ( $r$ , $d$ , a later term).

Common mistakes

Using the arithmetic condition for a geometric sequence
(or the reverse). Equal differences for arithmetic, equal ratios for geometric.
Writing $B^2 = A + C$
or $2B = AC$ . Say it in words: the middle term squared equals the product of its neighbours.
Dividing by $x$ or $d$ instead of factorising.
Dividing $x^2 = 3x$ by $x$ loses $x = 0$ ; dividing $6d^2 = 3ad$ by $d$ skips the step where $d \neq 0$ is used, which is where the reasoning is. The same goes for $225h^4 = 90h^5$ : factorise, then reject.

More mistakes (5)

Rejecting without a reason.
"Reject $-\frac{3}{2}$ " on its own loses the mark. Say which condition it breaks.
Leaving a rejected value in the answer
rejected values must go. In Example 2.1, leaving " $x = 6$ or $x = -\frac{3}{2}$ " as the final answer scores , even though $6$ is there.
Carrying $k$ from one part to the next
when the same terms are tested as arithmetic and then as geometric, the $k$ from one part is not a candidate in the other.
Skipping the expansion in a "show that"
multiply out every bracket on the page. Jumping straight to the given quadratic loses the .
Missing a zero term
if one of the three terms is 0, $B^2 = AC$ forces the middle term $B$ to be 0 too, and then one of the ratios is $\frac00$ , which is undefined. That value does not give a geometric sequence.

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