lesson
Multiply by the same factor
1 · Learn
Compare 3, 6, 9, 12 with 3, 6, 12, 24. The first adds 3 each time: it is arithmetic. The second multiplies by 2 each time: it is geometric. To test a geometric sequence with non-zero terms, divide each term by the term before it. A constant answer is the common ratio.
A ratio can be a fraction: 16, 8, 4, 2 repeatedly multiplies by 1/2. It can also be negative: 3, -6, 12, -24 multiplies by -2, alternating signs. A geometric sequence does not always grow.
If the first term is a and the common ratio is r, term n is a × r^(n - 1), where position n starts at 1 and ^ means a power. There are n - 1 multiplications after the first term. A few displayed terms do not uniquely prove a rule; use the rule stated in a task.
2 · Worked example
Start at 5 and multiply by 3 each time: 5, 15, 45, 135, 405. Term 5 is 5 × 3^4 = 405, not 5 × 3^5. The four gaps between five terms explain the exponent.
3 · Your turn
Start at 24 and multiply by 1/2 each time. Write the first five terms. Is their difference constant? Separately, start at 2 and multiply by -3; write four terms.
Check your answer
24, 12, 6, 3, 1.5. The differences are -12, -6, -3, -1.5, so they are not constant. The second sequence is 2, -6, 18, -54.
4 · Apply your learning
Make a position-and-term table for both practice sequences. Check consecutive ratios and compare them with consecutive differences. Explain why joining sequence points suggests non-integer positions that this model has not defined.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
Open your writing and drawing pad
Write, draw or show your working. Use a mouse, pen or finger. This pad is not a submitted answer.
Not saved to your account. Download your drawing as PNG and your typed notes as TXT before leaving or refreshing. They are separate files. Do not enter private information.
Open a picture saved on this device to draw on it again. Opening replaces the drawing, but Undo can restore it. PNG only, up to 4 MB and 4096 × 4096 pixels. The picture stays in this browser; it is not uploaded. Paper lines in a saved picture become part of that picture.
Clear the drawing? Your typed notes will stay. You can undo this action.
Your pad is empty.
Open a UTF-8 TXT file from this device, up to 40 KB and 10,000 characters. It stays in this browser. Opening can be undone until you type again. Downloads keep a new copy; they do not change the file you opened.
Replace the notes on this page with the opened file? Download the current notes first if you want to keep them.
Typed notes are not submitted answers.
Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
Practice this lesson
Lesson resources
Browse the available practice activities and worksheets for this lesson.
Your progress