Sequences and progressions

Foundations: the position is not the value

A sequence is a function defined at successive integer positions, for example n = 1, 2, 3, …. We write uₙ for the nth term. In the sequence 7, 11, 15, 19, …, we have u₁ = 7 and u₃ = 15. The index 3 is a position, whereas 15 is the value at that position; they are not two ways of writing the same number.

Some sequences start at n = 0. If we define vₙ = 5 × 2ⁿ for n = 0, 1, 2, …, the first value is v₀ = 5. If we start at n = 1, the first value of that same expression will be 10. Before using a formula, check where the index starts and what quantity the term represents.

A sequence graph plots points (n, uₙ). Inputs such as n = 2,5 do not automatically exist. Joining points with a line may help show a trend, but introduces intermediate values outside the original domain. A function of a real variable can provide an extension, as long as this new choice is acknowledged.

Explicit rules, recurrence and the limits of a pattern

An explicit rule calculates the term directly from the index. In uₙ = 4n + 3, n ≥ 1, we find u₁₀ = 43 without calculating the nine previous terms. A recurrence rule calculates from earlier terms and needs initial data: u₁ = 7 and uₙ₊₁ = uₙ + 4 for n ≥ 1 produce the same sequence.

The recurrence alone is not enough. The rule “add 4 to the previous term” can start at 7, at −2 or at another number, producing different sequences. A recurrence using two previous terms, such as uₙ₊₂ = uₙ₊₁ + uₙ, normally requires two initial values to fix the sequence.

Observing 2, 4, 6 does not prove that the next term is 8. That continuation is natural under the assumption of constant difference, but other rules coincide at the first three terms. For example, uₙ = 2n + (n − 1)(n − 2)(n − 3) gives 2, 4, 6 and then 14. We must distinguish “the proposed rule is” from “the few data points require”. Mathematical justification works with the rule, not just its appearance.

Arithmetic progression: growth through addition

In an arithmetic progression, or AP, the difference between consecutive terms is constant: uₙ₊₁ − uₙ = d. The number d is called the common difference. It can be positive, negative or zero. Starting from u₁, the second term is u₁ + d, the third is u₁ + 2d and, after n − 1 steps, the general term is:

The factor n − 1 counts the transitions between the first term and the nth term. Going from position 1 to position 6 takes five transitions, not six. If the index starts at zero, the corresponding formula is uₙ = u₀ + nd. This difference is a convention, not a contradiction.

If d > 0, the AP is strictly increasing; if d < 0, it is decreasing; if d = 0, it is constant. Between two terms, uⱼ − uᵢ = (j − i)d. Thus, knowing two distinct positions and their values allows us to calculate d. This is the structure of an affine function observed only at integer inputs, with the same increment at every step.

Sum an AP by pairing

Call Sₙ the sum of the first n terms. Write the sum once in normal order and once in reverse order. Adding corresponding sides, each pair equals u₁ + uₙ: the first and last, second and next-to-last, and so on. There are n pairs in this addition of two copies of the sum, so 2Sₙ = n(u₁ + uₙ).

Dividing by two corrects for counting the sum twice. The argument also works for odd n: presenting two copies avoids any difficulty with the central term. We do not need a separate rule for sums with an odd number of terms.

For example, one row of blocks may have 6 pieces, the next 9, then 12, continuing with a constant increase of 3. The sixth has 6 + 5 × 3 = 21 pieces. The total for all six rows is 6(6 + 21)/2 = 81 pieces. The term u₆ measures pieces in one row; S₆ measures pieces across all six rows. Confusing term and sum answers a different question.

Geometric progression: growth through a factor

In a geometric progression, or GP, each term is obtained by multiplying the previous one by a constant factor q: uₙ₊₁ = quₙ. With a nonzero first term and q ≠ 0, the quotient uₙ₊₁/uₙ confirms the common ratio. After n − 1 multiplications, uₙ = u₁qⁿ⁻¹. If q = 0, the recurrence produces u₁, 0, 0, …; handle this case directly to avoid dividing by zero terms.

A constant percentage gives a constant factor: increasing by 15% means multiplying by 1,15; decreasing by 15% means multiplying by 0,85. A percentage increase is not a constant absolute increase. Starting at 100, two increases of 15% give 100 × 1,15² = 132,25, because the second increase applies to 115.

For u₁ > 0, q > 1 produces growth; 0 < q < 1 produces a decrease towards positive values. If q < 0, the signs alternate, so “always increases” generally stops making sense. The magnitude depends on |q|. For example, with q = −1/2, signs alternate while magnitudes decrease. The condition on u₁ matters when asserting monotonicity.

Sum and compare multiplicative models

For a finite GP, Sₙ = u₁ + u₁q + … + u₁qⁿ⁻¹. Multiplying by q shifts the terms one position. Subtracting the two equalities cancels the intermediate terms and gives Sₙ − qSₙ = u₁ − u₁qⁿ. Therefore, if q ≠ 1, Sₙ = u₁(1 − qⁿ)/(1 − q). For q = 1, all terms are u₁ and Sₙ = nu₁.

An infinite sum is not obtained simply by deleting the last term. When |q| < 1, qⁿ approaches zero and the finite sums approach u₁/(1 − q). This value is the limit of the partial sums. Outside that condition, the formula generally does not give a finite infinite sum; the case where every term is zero is trivial and must be distinguished.

When comparing an AP with a GP, specify the horizon and parameters. A GP with ratio 0,9 decreases; it does not grow faster than an increasing AP. In applications, factors and increments may change over time. A regular model is useful within its assumptions; it does not guarantee that populations, prices or resources will follow the same rule forever.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The index n is the term's value.

n identifies the position; uₙ is the number associated with that position.

The formula starting from the first term contains n increments.

From position 1 to position n there are n − 1 transitions.

A constant percentage creates an AP.

It creates multiplication by a constant factor, that is, a GP while the conditions hold.

The geometric sum formula always allows infinitely many terms.

A nontrivial finite limit requires |q| < 1.