Sequences: from a verbal pattern to the nth term

Where we are on the map

Viewing a function only at integer steps creates an ordered sequence of values.

Begin with a familiar image

A staircase rising 3 cm each step has a constant difference; layers tripling each time have a constant factor.

Precise meaning

For arithmetic sequences, aₙ=a₁+(n−1)d; for geometric sequences, aₙ=a₁qⁿ⁻¹. The n−1 counts transitions from the first term.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

compare terms → difference or factor? → recursive rule → nth term → check

Worked example

5,8,11,14 has d=3 and aₙ=5+3(n−1). 2,6,18,54 has q=3 and aₙ=2×3ⁿ⁻¹.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Writing aₙ=a₁+nd counts n steps to the nth term, but reaching the first term requires zero transitions.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. For 12,17,22,... and 81,27,9,..., identify the type, write recursive and explicit rules, and find term 10.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The algebra checkpoint joins functions, systems, powers, polynomials, parabolas, growth, and sequences into one modeling chain.

Course contents