Natural numbers in a row form sequences. The simplest of these, the arithmetic progression, appears in many problems. Move the figure to see a PA grow:

Choose the first term and the reason. The bars are the terms; the sum comes out through the Gaussian trick.

A visual map of progressions

PA 2, 5, 8, 11, 14 of ratio 3.
Observe, predict and justify: what rule connects the terms?
Auditorium with increasing rows of reason 2.
A sequence helps you calculate everyday quantities.

Your mission: transform patterns into arguments

Discoveries itinerary

  1. Observe a sequence and identify how each term is obtained.
  2. Recognize a PA by the constant difference between consecutive terms.
  3. Relate the position of a term to its value using the general term.
  4. Justify the sum with pairs of terms and apply the idea in a concrete situation.

Start with introduction to progressions and proceed to arithmetic progression.

Connection with naturals: the positions of the terms are 1, 2, 3, … . The values ​​of a PA can also be negative or fractional; not all PAs have only natural terms. In the explorer, watch what happens when you change the ratio.

Challenge of the ranks

An auditorium has 12 rows. The first has 8 chairs and each following row has 2 more chairs. How many chairs are in the last row? What is the total capacity?

Check the answer and strategy

The last row has 8 + (12 − 1) × 2 = 30 chairs. The sum is 12 × (8 + 30) ÷ 2 = 228 seats. When joining the first row with the last row, each pair has 38 chairs; are 6 even.

Explain before using the formula

Why do we use n − 1 in the general term aₙ = a₁ + (n − 1)r?

See a justification

The first term is already in position 1. To get to position n, we take n − 1 steps, each one adding the ratio r.

Ready to complete? Create a PA of natural terms, calculate its tenth term and justify the sum of the first ten terms. Then, investigate what changes with zero or negative ratio.

Laboratory: recognize, calculate and check

In the training below, three missions appear: discover the reason, calculate a term and add terms. Write down your strategy before checking. The difficulty changes the number of terms considered. The questions in this activity use natural terms and ratios, including zero.

A round has 10 AP challenges. Use aₙ = a₁ + (n − 1)r and Sₙ = n(a₁ + aₙ)/2 when necessary.

Challenge: when do terms stop being natural?

A PA starts at 9 and has reason −2. Write the first six terms. Up to what position are all terms still natural?

Check out the investigation

The terms are 9, 7, 5, 3, 1 and −1. The first five are natural; the sixth is already negative. The position remains a positive natural, even when the value of the term is not natural.