Thirty-Six

Unit 5 · FunctionsF6 · Sequences · Guidance

Arithmetic & Geometric Sequences

Recognize constant differences and ratios, then find terms and finite sums.

  • 30 min
  • Unlocks: Staging Sequencer
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

common difference
the amount added between arithmetic terms

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Use differences for arithmetic sequences and ratios for geometric sequences. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Use differences for arithmetic sequences and ratios for geometric sequences. Keep exact values through the setup; round only when the stem asks.
Core relationship
an=a1+(n−1)da_n=a_1+(n-1)d

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

An explicit rule is efficient for a distant term, while a finite-sum formula is faster than adding every term. Desmos can evaluate the explicit rule, but first decide whether the question asks for one term or a sum.

Explore in Desmos

Evaluate the 18th arithmetic term

Define the explicit rule and evaluate a(18); algebra is faster for the finite sum.

The calculator is live: change anything and see what happens.

Evaluate the 18th arithmetic term

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

The multiplier is n−1n-1 because the first term requires zero jumps.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Find the 18th term of 7,11,15,…7,11,15,\ldots
The common difference is 4.

How many common-difference jumps occur before term 18?

a18=75a_{18}=75.

Why is the multiplier 17 rather than 18?

Worked example

Example 2: ACT twist

Problem

Find the sum of the first 12 terms of 5,8,11,…5,8,11,\ldots
The 12th term is 5+11(3)=385+11(3)=38.

Which values belong in the finite arithmetic-series formula?

S12=258S_{12}=258.

Why is the last term 38?

Faded example · you finish it

Example 3: You finish it

Problem

Find the 6th term of 3,6,12,24,…3,6,12,24,\ldots.
The common ratio is 2.

Complete the calculation.

Desmos way

Check the explicit sequence rule

Define the explicit rule and evaluate a(18); algebra is faster for the finite sum.

Check the explicit sequence rule

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.