Progressions Practice Questions

Q: How do I tell an arithmetic progression from a geometric one?

In an AP you add the same amount each time, a common difference ddd. In a GP you multiply by the same amount each time, a common ratio rrr. Check two or three gaps: equal differences mean AP, equal ratios mean GP.

Q: When can I use the sum to infinity formula?

Only when the geometric progression converges, which needs −1<r<1-1<r<1−1<r<1. Then S∞=a1−rS_{\infty}=\dfrac{a}{1-r}S∞=1−ra. If ∣r∣≥1|r|\ge 1∣r∣≥1 the terms do not shrink, so the sum grows without limit and the formula does not apply.

Q: How do I find a and d when I am only given two terms?

Write each term as an equation using Tn=a+(n−1)dT_{n}=a+(n-1)dTn=a+(n−1)d, then solve the pair simultaneously. Subtracting one equation from the other removes aaa and leaves you with ddd straight away.

Q: Do I lose all the marks if my final answer is wrong?

No. Because marking is analytic, a correctly quoted formula and a correct substitution still earn marks even if the arithmetic slips at the end. That is why you should always write the formula first.

How to use these practice questions

The six questions below rise in difficulty across the whole chapter, from the $n$ th term of an arithmetic progression to the sum to infinity of a geometric progression. Give yourself roughly five to eight minutes per question and work on paper first, writing every line the way you would in the real exam, the formula you are using, a clear substitution of $a$ , $d$ or $r$ , then the final answer.

Resist the urge to peek.

Only once you have committed to a full answer should you open the solution and mark yourself line by line. When your working differs from ours, stop at the exact line where the two part company; that single step is almost always where the mark was lost.

Because Add Math is marked analytically, a correctly quoted formula still earns credit even when the arithmetic slips, so always write the formula before you substitute.

Six practice questions

Q1[3 marks]

An arithmetic progression has first term $5$ and common difference $3$ . Find the $10$ th term.

Show worked solution

The $n$ th term of an AP is $T_{n}=a+(n-1)d$ . Write the formula, then substitute $a=5$ , $d=3$ and $n=10$ :

T_{10}=a+(10-1)d=5+9(3)

=5+27=32

Answer

The $10$ th term is $32$ . Quick check by listing: $5,8,11,14,17,20,23,26,29,32$ , the tenth value is indeed $32$ .

Q2[3 marks]

For the arithmetic progression $4, 7, 10, \ldots$ , find the sum of the first $12$ terms.

Show worked solution

Read off $a=4$ and $d=7-4=3$ . The sum of the first $n$ terms of an AP is $S_{n}=\dfrac{n}{2}\big[2a+(n-1)d\big]$ .

Substitute $n=12$ :

S_{12}=\frac{12}{2}\big[2(4)+(12-1)(3)\big]

=6\big[8+33\big]=6(41)=246

Answer

The sum is $246$ . Check with the other sum formula: the $12$ th term is $T_{12}=4+11(3)=37$ , so $S_{12}=\dfrac{12}{2}(4+37)=6(41)=246$ , which agrees.

Q3[4 marks]

The $3$ rd term of an arithmetic progression is $11$ and the $7$ th term is $27$ . Find (a) the first term and the common difference, and (b) the $20$ th term.

Show worked solution

(a) Turn each given term into an equation using $T_{n}=a+(n-1)d$ . From the $3$ rd and $7$ th terms:

a+2d=11 \qquad a+6d=27

Subtract the first equation from the second to eliminate $a$ , then back-substitute:

(a+6d)-(a+2d)=27-11 \;\Rightarrow\; 4d=16 \;\Rightarrow\; d=4

a+2(4)=11 \;\Rightarrow\; a=3

(b) Now use the $n$ th term formula with $a=3$ , $d=4$ , $n=20$ :

T_{20}=3+(20-1)(4)=3+76=79

Answer

$a=3$ , $d=4$ , and the $20$ th term is $79$ . Check the given terms: $T_{3}=3+2(4)=11$ and $T_{7}=3+6(4)=27$ , both correct.

Q4[4 marks]

A geometric progression has first term $3$ and second term $6$ . Find (a) the common ratio, and (b) the $6$ th term.

Show worked solution

(a) The common ratio is any term divided by the one before it, $r=\dfrac{T_{2}}{T_{1}}$ :

r=\frac{6}{3}=2

(b) The $n$ th term of a GP is $T_{n}=ar^{n-1}$ . Substitute $a=3$ , $r=2$ , $n=6$ :

T_{6}=3(2)^{6-1}=3(2)^{5}=3(32)=96

Answer

$r=2$ and the $6$ th term is $96$ . Quick check by listing: $3,6,12,24,48,96$ , the sixth value is $96$ .

Q5[4 marks]

For the geometric progression $2, 6, 18, \ldots$ , find the sum of the first $5$ terms.

Show worked solution

Read off $a=2$ and $r=\dfrac{6}{2}=3$ . Since $r>1$ , use $S_{n}=\dfrac{a(r^{n}-1)}{r-1}$ with $n=5$ :

S_{5}=\frac{2\big(3^{5}-1\big)}{3-1}=\frac{2(243-1)}{2}

=\frac{2(242)}{2}=242

Answer

The sum is $242$ . Check by adding directly: $2+6+18+54+162=242$ , which agrees.

Q6[5 marks]

A geometric progression has first term $18$ , and its sum to infinity is $27$ . Find (a) the common ratio, and (b) the third term.

Show worked solution

(a) The sum to infinity exists because the series converges, and $S_{\infty}=\dfrac{a}{1-r}$ . Substitute $a=18$ and $S_{\infty}=27$ , then solve for $r$ :

\frac{18}{1-r}=27 \;\Rightarrow\; 1-r=\frac{18}{27}=\frac{2}{3}

r=1-\frac{2}{3}=\frac{1}{3}

(b) With $a=18$ and $r=\dfrac{1}{3}$ , the third term is $T_{3}=ar^{2}$ :

T_{3}=18\left(\frac{1}{3}\right)^{2}=18\times\frac{1}{9}=2

Answer

$r=\dfrac{1}{3}$ and the third term is $2$ . Since $|r|<1$ , the sum to infinity is valid.

Check: $S_{\infty}=\dfrac{18}{1-\tfrac{1}{3}}=\dfrac{18}{\tfrac{2}{3}}=27$ , as given.

How to mark yourself like an examiner

Add Math is marked analytically, which means marks are attached to steps, not only to the final number. When you check your own script, award yourself credit the way a marker would: look for the correct formula, the right substitution, and a clean final statement.

Method mark: did you write the correct formula, $T_{n}=a+(n-1)d$ , $S_{n}=\dfrac{n}{2}[2a+(n-1)d]$ , $T_{n}=ar^{n-1}$ , or $S_{\infty}=\dfrac{a}{1-r}$ ?
Substitution mark: are $a$ , $d$ or $r$ and $n$ put into the correct places, with brackets kept around $(n-1)$ ?
Answer mark: is the final value stated clearly, and does it survive a check by listing terms or using a second formula?
For a sum to infinity, you must have $|r|<1$ for the answer to be valid, always confirm the ratio is a proper fraction.
If your final number is wrong but the formula and substitution are right, give yourself the method marks, that is exactly what a real marker does.

How a teacher helps

Marking yourself is powerful, but it is hard to see your own blind spots. In a one-to-one lesson our teacher watches the exact line where a mark slips away, the AP formula used for a GP, an $n-1$ written as $n$ , or a sum to infinity attempted when the ratio is larger than one, and corrects the habit on the spot.

Because our teachers are experienced, you work with someone who explains the why behind each formula. Lessons are taught in English, while SPM papers are set in both Malay and English, so we make sure the notation reads the same to you either way.

Get 1-to-1 help.

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Frequently asked questions

How do I tell an arithmetic progression from a geometric one?

In an AP you add the same amount each time, a common difference $d$ . In a GP you multiply by the same amount each time, a common ratio $r$ .

Check two or three gaps: equal differences mean AP, equal ratios mean GP.

When can I use the sum to infinity formula?

Only when the geometric progression converges, which needs $-1<r<1$ . Then $S_{\infty}=\dfrac{a}{1-r}$ .

If $|r|\ge 1$ the terms do not shrink, so the sum grows without limit and the formula does not apply.

How do I find

a

and

d

when I am only given two terms?

Write each term as an equation using $T_{n}=a+(n-1)d$ , then solve the pair simultaneously. Subtracting one equation from the other removes $a$ and leaves you with $d$ straight away.

Do I lose all the marks if my final answer is wrong?

No. Because marking is analytic, a correctly quoted formula and a correct substitution still earn marks even if the arithmetic slips at the end.

That is why you should always write the formula first.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

Progressions, Practice Questions

How to use these practice questions

Six practice questions

How to mark yourself like an examiner

How a teacher helps

Frequently asked questions

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