10.3
Suites arithmétiques
Une suite arithmétique est une suite ordonnée de nombres où chaque terme s’obtient en ajoutant au terme précédent une valeur constante, appelée raison…
Une suite arithmétique est une liste de nombres où chaque terme augmente ou diminue du même nombre fixe, connu sous le nom de différence commune. Considérez un tas de poteaux. La première couche contient 25 pôles, et le nombre de pôles continue de diminuer de 1 dans chaque couche successive.
Étant donné que le tas comporte 12 couches, l’objectif est de trouver le nombre total de poteaux.
Cette disposition forme une séquence arithmétique, car le nombre de pôles diminue d’une quantité constante d’une couche à l’autre.
Dans ce scénario, le nombre de pôles dans la 12e couche est calculé à l’aide de la formule du nième terme d’une suite arithmétique, en fonction du premier terme, de la différence commune et du nombre de couches. Les valeurs de ces termes sont ensuite substituées dans la formule, qui se simplifie en 25 moins 11, ce qui donne 14 pôles dans la 12e couche.
Le nombre total de pôles dans la pile, appelé somme partielle de la séquence, est ensuite calculé en prenant la moyenne du nombre de pôles dans la première et la dernière couche et en la multipliant par le nombre total de couches. On parle de somme partielle puisque seuls les 12 premiers termes de la suite sont additionnés. Il en résulte une somme partielle de 12 multipliée par la moyenne de 25 et 14, ce qui donne 234 pôles.
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Q1: What is a common difference in an arithmetic sequence?
The common difference is the fixed constant value added to or subtracted from each term to get the next term in an arithmetic sequence. In the pole pile example, the common difference is -1 because the number of poles decreases by one in each successive layer. This consistent pattern allows you to predict any term in the sequence without listing all preceding terms.
Q2: How do you find the nth term of an arithmetic sequence?
Use the formula aₙ = a + (n - 1)d, where aₙ is the nth term, a is the first term, d is the common difference, and n is the term position. For the pole pile with 25 poles in layer 1 and common difference -1, the 12th layer contains 25 + (12 - 1)(-1) = 14 poles. This formula eliminates the need to calculate every preceding term.
Q3: What is the partial sum formula for an arithmetic sequence?
The partial sum Sₙ equals n times the average of the first and last terms: Sₙ = n(a + aₙ)/2. For the 12-layer pole pile, the total is 12 × (25 + 14)/2 = 234 poles. This formula efficiently computes the sum of the first n terms without adding each term individually, making it practical for large sequences.
Q4: Why is the pole pile arrangement considered an arithmetic sequence?
The pole pile forms an arithmetic sequence because each layer contains a constant decrease of one pole from the previous layer. This consistent pattern of change—the common difference—is the defining characteristic of arithmetic sequences. The structured arrangement allows systematic calculation of any layer's pole count and the total across all layers.
Q5: How does the partial sum differ from the total sum of an infinite sequence?
A partial sum calculates only the first n terms of a sequence, as shown when finding the total of 12 layers in the pole pile. The partial sum formula Sₙ = n(a + aₙ)/2 applies to a finite number of terms. This approach is practical for real-world applications where you need the cumulative total of a specific portion rather than an entire infinite sequence.
Q6: Can you use arithmetic sequences to solve real-world stacking problems?
Yes, arithmetic sequences model real-world stacking scenarios like the pole pile, where items decrease or increase by a constant amount per layer. By identifying the first term, common difference, and number of layers, you can calculate individual layer quantities using the nth term formula and total inventory using summation notation. This systematic approach simplifies inventory and structural calculations.
Q7: What is the relationship between arithmetic sequences and geometric sequences?
Arithmetic sequences use addition of a constant common difference, while geometric sequences use multiplication by a constant ratio. Both are structured numerical patterns that allow efficient term and sum calculations. Understanding arithmetic sequences provides foundational knowledge for comparing with geometric sequences and recognizing which pattern applies to different real-world or theoretical problems.