Geometric Progressions

Pure Mathematics 1

Mathematics9709

Multiply by the same number each time

A (GP) uses a constant multiplier, the common ratio $r$ . Divide a term by the previous non-zero term to find it.

Starting at the first term $a$ , term $n$ needs $n-1$ multiplications:

Term 1a = 8
↓ multiply by r = −½
Term 2ar = −4
↓ multiply by r = −½
Term 3ar² = 2

Multiply by −½: the sign changes and the size halves.

u_n=ar^{n-1}

The magnitude $|r|$ is the size of the ratio without its sign. If $0<|r|<1$ , term sizes shrink; if $|r|>1$ , they grow. A negative ratio alternates signs. With $r=1$ terms are constant; with $r=0$ all terms after the first are zero.

If $u_n=9(-2/3)^n$ , substitute $n=1$ : $a=-6$ , not 9. Each extra power still multiplies by $r=-2/3$ .

Use the gap between known terms

Worked example

A GP has $u_2=12$ and $u_5=40.5$ . Find its first term and ratio.

$ar=12$ and $ar^4=40.5$ . The first equation is non-zero, so division is allowed:

r^3=\frac{40.5}{12}=\frac{27}{8}

The cube root reverses cubing: $(3/2)^3=27/8$ , so $r=3/2$ . Then $a=12/(3/2)=8$ . The exponent 3 counts the three multiplications from term 2 to term 5.

An even power keeps two possible signs: $r^2=9$ gives $r=3$ or $r=-3$ . Find the matching $a$ for each ratio. Reject a branch only when a given condition rules it out.

Now you try

Your turn9709/12 Feb/Mar 2023 Q4(b) [3 marks]

A tree trunk has circumference 5.00 m, then 5.02 m one year later. Given instead that the yearly measurements form a GP, find its circumference 20 years after the first measurement.

Show worked answer

The ratio is $5.02/5=1.004$ . There are 20 multiplications:

\begin{aligned}u_{21}&=5(1.004)^{20}\\&=5.415571\ldots\end{aligned}

Answer

5.42\text{ m}

Keep the unrounded calculator value until the final 3 significant figures.

Your turnIndependent practice

A GP has $u_2=6$ and $u_4=24$ . Find all possible first terms and ratios.

Show worked answer

$r^2=24/6=4$ . If $r=2$ , $a=3$ . If $r=-2$ , $a=-3$ . Both pairs give the stated terms.

Multiply by the same number each time

A (GP) uses a constant multiplier, the common ratio $r$ . Divide a term by the previous non-zero term to find it.

Starting at the first term $a$ , term $n$ needs $n-1$ multiplications:

Term 1a = 8
↓ multiply by r = −½
Term 2ar = −4
↓ multiply by r = −½
Term 3ar² = 2

Multiply by −½: the sign changes and the size halves.

u_n=ar^{n-1}

If $u_n=9(-2/3)^n$ , substitute $n=1$ : $a=-6$ , not 9. Each extra power still multiplies by $r=-2/3$ .

Use the gap between known terms

Worked example

A GP has $u_2=12$ and $u_5=40.5$ . Find its first term and ratio.

$ar=12$ and $ar^4=40.5$ . The first equation is non-zero, so division is allowed:

r^3=\frac{40.5}{12}=\frac{27}{8}

The cube root reverses cubing: $(3/2)^3=27/8$ , so $r=3/2$ . Then $a=12/(3/2)=8$ . The exponent 3 counts the three multiplications from term 2 to term 5.

An even power keeps two possible signs: $r^2=9$ gives $r=3$ or $r=-3$ . Find the matching $a$ for each ratio. Reject a branch only when a given condition rules it out.

Now you try

Your turn9709/12 Feb/Mar 2023 Q4(b) [3 marks]

A tree trunk has circumference 5.00 m, then 5.02 m one year later. Given instead that the yearly measurements form a GP, find its circumference 20 years after the first measurement.

Show worked answer

The ratio is $5.02/5=1.004$ . There are 20 multiplications:

\begin{aligned}u_{21}&=5(1.004)^{20}\\&=5.415571\ldots\end{aligned}

Answer

5.42\text{ m}

Keep the unrounded calculator value until the final 3 significant figures.

Your turnIndependent practice

A GP has $u_2=6$ and $u_4=24$ . Find all possible first terms and ratios.

Show worked answer

$r^2=24/6=4$ . If $r=2$ , $a=3$ . If $r=-2$ , $a=-3$ . Both pairs give the stated terms.

Multiply by the same number each time

Use the gap between known terms

A percentage change is a multiplier

Now you try

Geometric Progressions

Multiply by the same number each time

Use the gap between known terms

A percentage change is a multiplier

Now you try