Running and confirming an iteration | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Use Ans to repeat the formula

If the formula contains a trigonometric or inverse-trigonometric function, use the angle unit required by the question. An interval written using $\pi$ means the calculator must be in radian mode.
Enter the given starting value $x_0$ and press equals so it becomes the current calculator answer. The calculator label Ans means this previous, internally stored answer.
Enter $F(\mathrm{Ans})$ and press equals. This is $x_1$ .
Record $x_1$ at the requested display precision, then press equals again for $x_2$ . Continue in the same way.

A calculator table or recurrence feature is also valid if it applies the correct rule. For a short exam sequence, the Ans method is usually the fastest.

Common mistake

Do not type a rounded written value back into the formula. Ans keeps more internal digits and prevents rounding error from accumulating.

Show enough terms to support the answer

Worked example

Run the logarithmic sequence from x₀ = 0.45

Use $x_{n+1}=\tfrac12\ln(3-x_n)$ and give each iteration to 5 significant figures. Decimal places count digits after the decimal point; significant figures start at the first non-zero digit.

Value	5 significant figures
$x_0$	$0.45000$
$x_1$	$0.46805$
$x_2$	$0.46450$
$x_3$	$0.46520$
$x_4$	$0.46506$
$x_5$	$0.46509$
$x_6$	$0.46508$
$x_7$	$0.46508$

The terms alternate, but the gap between them shrinks. The last two both give $0.46508$ to the requested 5 significant figures, and the sequence supports $x=0.465$ to 3 significant figures. (9709/22/O/N/24 Q4(c))

Use the exact starting value in the question.
Write every requested iteration, not only the final root.
The iteration terms and final answer can be requested at different precisions. Follow both instructions separately.
Do not stop because two heavily rounded values look equal. Continue until the displayed evidence is stable enough for the final precision.

Check the complete rounding interval

The rounding interval for 0.465

0.4645 ≤ α < 0.4655

A sign change at the two boundaries brackets the root inside this interval.

To confirm 0.465, bracket the root inside its rounding interval.

A value that rounds to $0.465$ to 3 significant figures lies from $0.4645$ up to, but not including, $0.4655$ . These are the two halfway boundaries.

For the original equation, use the following continuous function:

f(x)=1+e^{2x}-(4-x)

f(0.4645)=-0.003524\ldots

f(0.4655)=0.002545\ldots

The sign change places the root inside the complete rounding interval, so $0.465$ is confirmed to 3 significant figures.

Key idea

Sufficient iteration terms can justify a rounded root. A sign change across its two rounding boundaries is a direct independent check.

Your turnIndependent check [3 marks]

Starting with $x_0=1$ , use $x_{n+1}=\ln(4+x_n)$ . Give each iteration to 5 significant figures and the root to 3 significant figures.

Show worked answer

x_0=1.0000

x_1=1.6094,\quad x_2=1.7245

x_3=1.7447,\quad x_4=1.7483

x_5=1.7489,\quad x_6=1.7490

x_7=1.7490

Answer

x=1.75\quad(3\text{ s.f.})