Problems with two progressions

6 minAS Level1.6.21.6.31.6.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can solve problems that involve an arithmetic and a geometric progression at the same time, by naming the unknowns and turning each sentence into an equation.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$u_n$	the $n$ th term
$a$ , $d$	the first term and common difference of the AP (arithmetic progression: add $d$ each time)
$r$	the common ratio of the GP (geometric progression: multiply by $r$ each time)
$S_\infty$	the sum to infinity of a convergent GP, $\dfrac{a}{1 - r}$

The crossover episode

Cambridge loves a crossover: the same numbers are terms of an AP and a GP.

It looks scary. It isn't, if you follow a routine:

Name the unknowns: usually $a$ and $d$ for the AP, $r$ for the GP.
Translate every fact into an equation, using $u_n = a + (n - 1)d$ and (middle)² = product of the outer two.
Solve, then use any condition like " $d \neq 0$ " or " $|r| < 1$ " to reject roots.

The translation table

The question says…	You write
the 3rd term of the AP	$a + 2d$
the 1st, 3rd and 7th terms of an AP are in GP	$(a + 2d)^2 = a(a + 6d)$
$p$ , $q$ , $s$ are in AP	$2q = p + s$
$p$ , $q$ , $s$ are in GP	$q^2 = ps$
the GP is convergent	$\lvert r \rvert < 1$

The key move is the second row: write each term of the AP in terms of $a$ and $d$ , then apply the GP condition to those expressions.

Worked example: Terms of an AP that form a GP

An arithmetic progression has first term 3 and common difference $d$ , where $d \neq 0$ . Its 1st, 2nd and 5th terms are the first three terms of a geometric progression. Find $d$ and the common ratio of the geometric progression.

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Quick check

Multiple choiceQuestion 1 · Recall

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