How do you check whether a sequence is geometric?

Divide each term by the one before it and see whether you always get the same number. If that ratio changes anywhere, the sequence is not geometric.

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Geometric Sequences & Series (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

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FrontGeometric Sequences
Define common ratio.

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Cards in this collection (11)

Define common ratio.
The common ratio $r$ is the fixed number each term of a geometric sequence is multiplied by to get the next one.
It is found by dividing any term by the term before it.
The $n$ th term of a geometric sequence is:
$u_{n} = a r^{______}$
$u_{n} = a r^{n - 1}$
The power is $n - 1$ rather than $n$ because the first term has been multiplied by $r$ no times yet.
You know the 3rd and 7th terms of a geometric sequence. How do you find $a$ and $r$ ?
Write both using $u_{n} = a r^{n - 1}$ , then divide one equation by the other.
The $a$ cancels, leaving a single equation in $r$ ; substituting back then gives $a$ .
True or False?
The terms of a geometric sequence always get bigger.
False.
If $|r| < 1$ the terms shrink towards zero, and a negative $r$ makes them alternate in sign.
Only $r > 1$ gives a sequence that grows steadily.
How do you check whether a sequence is geometric?
Divide each term by the one before it and see whether you always get the same number.
If that ratio changes anywhere, the sequence is not geometric.
The sum of the first $n$ terms of a geometric series is:
$S_{n} = \frac{a (1 - r^{n})}{______}$
$S_{n} = \frac{a (1 - r^{n})}{1 - r}$
The equivalent form $\frac{a (r^{n} - 1)}{r - 1}$ is more convenient when $r > 1$ , because it keeps everything positive.
How is the formula for a geometric series proved?
Write the sum out, write $r$ times the sum underneath it, and subtract.
All but two terms cancel, leaving $S_{n} - r S_{n} = a - a r^{n}$ , which factorises and rearranges into the formula.
In the proof of the geometric series formula, why do you multiply by $r$ before subtracting?
Because multiplying by $r$ shifts every term along by one place, so nearly all of them line up with a copy of themselves.
Subtracting then wipes out everything except the first term and the last new one.
What is the sum to infinity of a geometric series, and when does it exist?
$S_{\infty} = \frac{a}{1 - r}$ , and it exists only when $|r| < 1$ .
Otherwise the series is divergent and has no sum to infinity at all.
True or False?
A geometric series with $r = - 0.5$ has a sum to infinity.
True.
The condition is on the modulus of $r$ , and $|- 0.5| = 0.5$ , which is less than $1$ .
The terms alternate in sign but still shrink towards zero, so the total settles on a finite value.
Why might logarithms be needed in a geometric series question?
Because $n$ appears as a power, in $r^{n}$ .
Asking how many terms are needed to reach a given total therefore means solving an exponential equation.

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