Arithmetic sequences — Mathematics, 14–17
How a steady change creates a sequence, and how to find any term without listing all the earlier ones.
The same step each time
An arithmetic sequence changes by the same amount whenever you move from one term to the next. For example, 7, 11, 15, 19 goes up by 4 each time. The first term and this constant change are enough to describe the whole sequence.
Why use a formula?
Listing a sequence is easy at the start but slow when someone asks for the 1,000th term. The formula aₙ = a₁ + (n − 1)d counts how many equal steps are needed. It came from replacing repeated addition with one calculation.
Finding a far-away term
A cinema row has seat numbers 12, 15, 18, 21 and so on. Here a₁ = 12 and d = 3. For the 20th seat, calculate a₂₀ = 12 + (20 − 1) × 3 = 12 + 57 = 69. The pattern predicts seat 69 without listing the other 16 numbers.
The index is not the value
A common mistake is to use n × d instead of (n − 1)d. It feels natural because the 20th term sounds like 20 steps, but the first term is already present before any step is taken. From term 1 to term 20 there are only 19 moves.
Steady plans
This model fits situations with a fixed increase or decrease: seats in a theatre row, monthly savings that rise by the same amount, or steps between marked points. It is not suitable when the change itself grows, such as interest that compounds. Checking the pattern first matters.
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